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Lattices Of Modal Logics

Frank Wolter

Dissertation am Fachbereich Mathematik der Freien Universität Berlin

Eingereicht im August 1993

Betreuer der Dissertation

Prof. Dr. W. Rautenberg

1

Contents

1 Basic Concepts 10

1.1 Syntax............................................................................................................ 10

1.2 Modal Algebras.............................................................................................. 11

1.3 Generalized Frames........................................................................................ 12

1.4 Completeness and Persistence....................................................................... 13

1.5 Polymodal Logics........................................................................................... 15

1.5.1 Fusions .............................................................................................. 15

1.5.2 What is the upper part of A / j? ......................................................... 16

2 Sublattices of Mn 20

2.1 Describable Operations...................................................................................... 20

2.2 Subframe Logics ............................................................................................... 23

2.2.1 Exam ples............................................................................................... 25

2.3 Confinal Subframe Logics and other examples................................................. 26

2.4 Splittings in Modal L o g i c .................................................................................27

2.4.1 Sp littings............................................................................................... 27

2.4.2 A General Splitting-Theorem................................................................29

2.4.3 A Counterexample................................................................................. 33

2.4.4 The Use of Splittings.............................................................................. 34

2.4.5 Some Results of W .J.B lok...................................................................... 39

3 Subframe Logics 40

3.1 General Properties............................................................................................. 40

3.1.1 S/-splitting-formulas.............................................................................. 42

3.1.2 Subframe Logics and Confinal Subframe Logics above K4 .................. 44

2 CONTENTS3.2 Basic Splittings of SNfn ...............................................................................41)

4 The Lattice of Monomodal Subframe Logics 54

4.1 Basic Monomodal Splittings.......................................................................... 54

4.2 Subframe Logics above K4 ( I I ) .................................................................... 56

4.3 A Chain of incomplete Subframe L o g ic s ...................................................... 58

4.4 Subframe Logics above K4 ( I I I ) .................................................................... 61

4.5 Simple incomplete and not strictly 5/-complete L og ics ...................................66

4.6 A note on Neighbourhood Semantics............................................................... 69

4.7 Subframe spectra...............................................................................................69

5 5/-completeness in Polymodal Logic 71

5.1 Connected Logics............................................................................................... 72

5.2 5/-splittings in Lattices of Connected Logics ...................................................73

5.3 The upper part of SAfn ....................................................................................78

5.4 Tense Logics .....................................................................................................84

5.4.1 Descendants and Variants......................................................................87

5.4.2 Subframe logics above K4 ( I V ) ..........................................................95

5.4.3 Proof of Theorem 5.4.2 ..................................................................... 96

5.4.4 Some Remarks on the F M P ............................................................... 106

6 Splittings and ^/-splittings in some sublattices of A/j 109

7 Ä-persistent Subframe Logics without the FM P 113

8 Index 116

9 List of Symbols 118

CONTENTS 3

10 List o f Logics 119

11 References 120

12 German Summary 123

13 Curriculum Vitae 124

4 INTRODUCTIONIntroduction

In this paper we investigate modal logics from a lattice theoretic point of view. There are essentially two well-known methods of research on lattices of modal logics. The investigation of the lattice of modal logics as a whole, and the local investigation of the lattice of extensions of some strong logic. In this thesis we will provide a third one by investigating proper complete sublattices, which are not filters within the whole lattice, but compactness preserving. The framework of our investigation is the lattice of normal n-modal logics (i.e. normal modal logics with n modal operators), which is denoted by Afn. The lattice of normal monomodal logics is also denoted by Af.

It is now more than one decade ago that the lattice theoretic point of view on modal logics filled as much papers as for instance completeness theory or correspondence theory. Perhaps the first non-trivial, explicite lattice theoretic theorem was the result of Makinson [71] that the lattice of normal monomodal logics has exactly two co-atoms. It followed a discussion of the lattice of extensions of K4. The first results were obtained by Maksimova [75b], Maksimova & Rybakow [74] and Blok [76]. Maksimova [75a] and Esakia & M eskhi [77] independently proved that there are exactly 5 pretabular extensions of 54. Rauten berg [79],[80] used the technique of splittings to give lattice-theoretic descriptions of most of the standard systems above K 4. Blok [80a] showed that a logic above K 4 is tabular if and only if it has finite codimension. In this paper he also proved that there exist 2N° pretabular logics above /if4, contrasting to the situation above 54.

At about the same time systematic research on the lattice Af started. Rauten BERG [77] observed that the lattice Af is distributive and used JÖNSSONS Lemma for proving that tabular logics are finitely axiomatizable and that extensions of a tabular logic are tabular again. It was discovered that the upper part of Af is more complicated than the upper part of the lattice of extensions of K 4. Blok [78] showed that both of the coatoms have 2h° lower covers, among which are 2H° incomplete logics. His research on the lattice of normal modal logics culminated in the characterization of the strictly complete logics and the result that a logic is either strictly complete or has degree of incompleteness 2h°. If a logic is not strictly complete, then it has 2K° lower covers. These deep results had one disadvantage: They are negative because they show that all standard systems with the exception of Z f(O T) are not strictly complete and have no interesting positive lattice theoretic property within Af. Thus, for Af lattice theoretic methods did not provide positive results, contrasting to the situation above K4.

In the following years and up to now research on lattice theoretic questions is restricted to the lattice of extensions SA of some strong logic A. The main example is of course the lattice of extensions of K 4. In Kracht [93] splittings of £K4 are used to prove the fmp for standard systems above K4. The concept of a canonical formula, introduced in Zakharyaschev [87],[92], coincides in the standard cases with splitting-formulas for sublattices of £K4, as will be shown in this thesis. Nevertheless, I think that it is correct to say that nowadays most of the interesting questions above K4 lie outside the scope of lattice-theoretic notions. Outside £K4 there are investigations from Nagle & T homason

INTRODUCTION 5[85] about the lattice of extensions of Kb and from Segerberg [86] about the extensions of K.Alh.

It is obvious that most of the negative results in monomodal logic transfer to lattices of polymodal logics. The situation is even worse. For instance the mentioned result of Makinson does not hold for polymodal logics. It is well known that even the lattice of tense logics has infinitely many incomplete co-atoms. Also the tool of splittings, which is basic for studying extensions of K 4, is not directly applicable to polymodal logics. For instance, K racht [92] shows that the lattice of extensions of the minimal tense extension of A'4 has only the trivial splitting. This situation seems to be one reason for the fact that the interest in polymodal logic lies mainly in definability theory or the investigation of rather specific systems.

The aim of this dissertation is to show that W . J. B lok ’s results and the negative results about lattices of polymodal logic do not force us to restrict lattice theoretic investigations to lattices of extensions of strong logics. It follows from these results that one cannot formulate interesting and solvable lattice theoretic problems by referring only to the lattice of normal (poly-) modal logics as a whole. But this difficulty is manageable by looking at some proper complete sublattices. Since we do not want to leave the whole lattice out of sight we restrict attention to sublattices D for which for a finitely axiomatizable logic A its upward projection A t ° : = f ] { ^ / 2 A|A' G D} is finitely axiomatizable as well. Such lattices are called compactness preserving sublattices of Afu. In Chapter 2 some general properties of such lattices are proved and a characterization of splittings is given. Examples are the lattices of n-modal subframe logics, denoted by SAfn, and the lattice of confinal subframe logics above K4, which was introduced in Zakharyaschev [92]. Subframe logics above K4 were introduced in F ine [85]. There is no obvious way to extend K. F ine ’s definition to non-transitive modal logic. However, a rather natural definition is as follows: Consider a (generalized) frame Q = (g, <1, A). Then, for each b € A the structure (6, < fl (b x 6), {o fl 6|a € A } ) is a frame again, and we call it a subframe of Q. Define an operation Sf on the class of frames by

Sf{Q) := {H\H a subframe of (7).A logic A is a subframe logic if the class of A-frames is closed under Sf. Lattices of subframe logics are the main subject of this thesis and we first discuss some reasons for this choice.

(1) The n-modal subframe logics form a compactness preserving sublattice of Afn. Moreover, given an axiomatization of a logic A, the upward projection of A, denoted by A , can be axiomatized effectively. The lattice of monomodal subframe logics lies rather natural within M. For instance, it will be shown that

K { O T ) f ! = T = f f (D p - * p ) ,

A '(D O p—*• ODp) = Grz (Grz = Grzegorczyks-system),

A'(ü± V OüJ.) = G (G = Gödel/Löb-system).

(2) The subframe logics above K4 are precisely the subframe logics defined by K. Fine; it will be shown that a complete logic is a subframe logic iff its Kripke frames are closed

6 INTRODUCTIONunder arbitrary substructures. About half of the systems discussed in the literature on modal logic are subframe logics.

(3) Forming the subframes of a frame is a natural operation. Short reflection shows that subframes correspond to the relativizations of the boolean reduct of a modal algebra or, more general, subframes correspond to the relativizations of cylindric algebras defined by Henkin, Monk & Tarski in [71]. This means on the syntactical side that a logic is a subframe logic iff its tautologies are closed under relativization to any proposition. (The relativization of a formula <j> to a propositional variable is defined in Chapter 2.2 and is called the S/-formula of <f>). This seems to be a natural condition in many intensional contexts, e.g. for □ as necessity or □ as action. Another possible area of application is □ as provability. Indeed, all known provability logics turn out to be subframe logics. For instance, the provability logic of Peano arithmetic coincides with G. At present the important provability logic of bounded arithmetic, TAo + f l i , is not known. But it follows immediately from the results of BERARDUCCI & VERBRUGGE [93] that this logic is a subframe logic iff it coincides with G.

In Fine [85] it is shown that all subframe logics above A'4 have the finite model property. The question whether this result extends to all subframe logics was one of the issues of this paper. It turned out that this is not the case and that, to the contrary, there are quite a lot of incomplete subframe logics with simple axiomatizations. One central part of this paper deals with the location of complete and incomplete subframe logics within the lattice of n-modal subframe logics. The other central part, strongly connected with the first one, is the study of splittings in lattices of subframe logics. Thus, we look at subframe logics from a lattice theoretic point of view.

The dissertation is divided into three parts. In the first part some basic concepts of modal logic are introduced. The concept of a complex variety is defined, which is the main subject of Goldblatt [89]. It is shown that the variety of modal algebras corresponding to a modal logic is complex if and only if the logic is compact. This theorem connects a purely algebraic concept with the model theoretic concept of compactness and prepares the investigation of general properties of subframe logics in Chapter 3. It was mentioned before that a lot of literature on monomodal logics investigates the upper part of the lattice M. Now one may ask why to start an investigation of lattices of polymodal logics with subframe logics and not with upper parts of lattices of polymodal logics. The answer is simple. As a rule, an interesting filter even in the lattice of bimodal logics is as complicated as the whole lattice of monomodal logics and all the negative results transfer to these lattices. This is shown in Section 1.5.2 by means of an embedding of the lattice of extensions of T into the lattice of extension of a bimodal logic S5 & A, where A is a quite simple tabular logic. (A i # A2 denotes the fusion of A i and A2, introduced in Kracht & W olter [91].)

The second part introduces the notion of compactness preserving sublattices of Nfn, c.p. sublattices for short. The concept of a describable operation is defined and used to give a model-theoretical characterization of c.p. sublattices. A describable operation is a closure operation C on the class of frames (or, equivalently, the class of modal algebras)

INTRODUCTION 7such that for a formula <f> there is a formula <f>c with C{Q) ^ <f> iff Q (= <ff for all frames Q. Now call a logic a C-logic if its frames are closed under C. It will be shown that a subset of Afn is a c.p. sublattice of Afn iff it is the set of C-logics for a describable operation C. Forming the subframes of a frame, that is, the operation Sf, defines a describable operation. Another example is the forming of confinal subframes in the sense of M. Zakharyaschev. After that we give a model-theoretic and syntactical characterization of splittings in c.p. sublattices by using the notion of a describable operation. From this characterization the main properties of the

splitting-formulas (alias Jankov-formulas or frame-formulas),

subframe-formulas (F ine [85]) and

canonical formulas for confinal subframe logics (Z akharyaschev [92])

follow as easy corollaries. In fact, all these formulas describe splittings in different lattices of modal logics and therefore are in a sense instantiations of the theory of splittings initiated by McK enzie [72]. The proof of the splitting-theorem is inspired by the investigation of splittings of filters in N n in Kracht [90a]. The advantage of the proof given here is that we get Jonssons Lemma (for modal varieties) as a corollary and that we do not restrict attention to finitely presentable algebras. 1 believe that the proof of Jonssons Lemma for varieties of modal algebras is of some relevance, for it is of model-theoretic nature and uses only methods known from other areas of modal logic. Part 2 is pursued with some propositions concerning the use of splittings in modal logic. Here the notion of Fine-spectra is relativized to lattices of subframe logics. The 5/-Fine-spectrum of a subframe logic 0 € SA := SAfn H SA is Fns^iß) := {0 i € «SA|.Fr(©) = / Y (0 i ) } . The ^/-degree of incompleteness of 0 above A is the cardinality of Fn$a (0 ) and 0 is strictly 5/-complete above A if this degree is 1. The notion of 5/-degree of incompleteness will be the main tool to locate the complete and incomplete subframe logics. The notions of splittings and strict completeness are connected by the observation that a complete logic which is an iterated splitting of SA by tabular logics, by finite frames for short, is strictly 5/-complete above A.

One reason for the investigations in Part 2 was the observation that the well-known splitting-formulas and subframe-formulas are two sides of the same coin. I believe that this observation can even be useful in the general context of investigations of lattices of subvarieties of a given, not necessarily modal, variety. One reason that splittings occur rather seldom in the literature seems to be the fact that in many cases there are simply not enough splittings of the whole lattice. The idea to overcome this problem is to investigate splittings in proper sublattices. Most of the concepts and proofs of this chapter have a straightforward translation into a more general context.

The third part investigates lattices of monomodal and polymodal subframe logics. In Chapter 3.1 a theorem of K. F ine is extended by showing that for subframe logics the concepts of fi-persistence (or of a natural logic), canonicity, compactness, and complex varieties are equivalent. Note that this does not hold for all modal logics because there are canonical logics which are not ^-persistent.

8 INTRODUCTIONThe chapters 3.2 - 6 hinge on the notions of S/-degree of incompleteness and splittings.

There are mainly three problems we try to solve:

( 1) Given a subframe logic. A, characterize the logics which split SA.

(2) Given a logic 0 € SA. Decide whether 0 is an iterated splitting of SA by finite frames.

(3) Given a logic 0 € SA. Decide whether 0 is strictly 5/-complete above A.

It will follow from the splitting theorem that problem (1) causes no troubles for intransitive logics A with the finite model property, since in this case a logic 0 splits SA if and only if 0 = Th(Sf{g)) for a finite rooted frame g. Chapter 6 and parts of Chapter 3 are concerned with problem ( 1) in the non m-transitive case. A complete answer is given for the lattices SAfn and SA with A = # nT, K.t, 55 #5 5 and A 4 # A4. It turns out that for Afn and £ (K 4 # K4) there is a one to one correspondence between splittings of Afn and splittings of SAfn, respectively between splittings of £ (A 4 # K4) and of 5 (A 4 # A'4). In the other three cases the situation is completely different. The lattices £K.t, £ (# nT ) and 5(55 # 55) have only the trivial splittings but the corresponding lattices of subframe logics have many splittings with nice properties.

In Chapter 3, 4 and 5 iterated 5/-splittings are used to determine classes of strictly 5/-complete logics and several classes of frames are defined to determine incomplete and not strictly 5/-complete subframe logics. Here, among others, the strictly 5/-c.omplete monomodal standard-systems are determined. For instance, it turns out that a subframe logic above K 4 is strictly 5/-complete, if and only if it is not weaker than G’ .3. It is shown that contrary to the situation in AT there is a logic in SAf which has degree of incompleteness 2**° in SAf but has only finitely many lower covers in SAf. Surprisingly enough, the logic G is such a logic with exactly 3 lower covers in SAf.

In the case of polymodal logic we mainly investigate lattices of connected logics. The extensions of K.t, where K.t denotes the minimal tense extension of K, and of 55 #55 are examples of connected logics. It is proved that in these lattices there exist large classes of strictly 5/-complete logics. The perhaps most surprising result is that K4.t is strictly complete in SK.t, contrasting to the result that K 4 has degree of incompleteness 2M° in SAf. It follows immediately that there exist 2K° monomodal subframe logics whose minimal tense extension is equal to K4.t. Another result, again contrasting to the situation above A4, is that for a monomodal subframe A above K 4 its minimal tense extension A.t is strictly complete in SK.t if and only if the class of A-frames is elementary. These results show that there are quite a lot of surprising phenomena in lattices of subframe logics and that we have a completely different situation compared to the lattices Afn.

In Chapter 5.3 the upper part o f the lattices SAfn is investigated. The main result is that for all n,/ all logics in 5 (# nA.A/f/) are strictly 5/-complete. In this case I have not been able to give a proof via splittings, hence a direct proof is delivered. It follows immediately that a subframe logic has finite codimension in SAfn if and only if it is tabular.

INTRODUCTION 9Thus, the situation in the upper part of lattices of subframe logics is comparable with the upper part of SK4. Among the pretabular logics in SAf there is only one not strictly .S/-complete logic, namely G.3. But even for 6'.3 the situation is not too complicated since G.3 has only 4 lower covers in SAf.

Questions concerning the fmp and decidability of subframe logics are merely touched upon in this paper. In Section 5.4.4 the techniques introduced for the determination of splittings are used to show that a wide range of non-elementary and natural subframe logics above K4.t do not have the fmp. In Chapter 7 we give an example of a monomodal, complete and elementary subframe logic without fmp. These examples and a bimodal example given in Wolter [93] show that we should not expect too many positive results concerning the fmp for subframe logics not above K4. Nevertheless, some general and positive results as regards the fmp of subframe logics above K4.t can be obtained. These results are stated in Section 5.4.4 without proofs. The proofs will be published elsewhere, since the methods involved are different from those introduced in this paper.

Throughout this paper the following notations will be used. For a set A the cardinality of A is denoted by \A\. If (g, < ) is a frame, we draw g in such a way that • represents a reflexive point and x represents an irreflexive point. A completed proof is indicated by H.

Several persons contributed to this thesis. The first to thank is Prof. W. Rauten berg for his supervision and generous support. Deeply indebted I am to M. Kracht for both his inspiring ideas when I started to write this thesis as well as for several dicussions during the last two years. He did not only teach me how to use splittings in modal logic but also insisted that the subframe logics defined in Fine [85] should be generalized to modal logics not above K4. In addition, I wish to thank M. Zakharyaschev for his advice on logics above K4 during my visit in Moscow. His ideas and results concerning elementarity of logics above K4 heavily influenced the investigation of strict ^/-completeness for tense logics. Many thanks also to A. Bull, C. G refe, V. Rybakow and V. Shehtman. Last but not least I thank I. W ulff for moral as well as technical support.

10 1 BASIC CONCEPTS1 Basic Concepts

1.1 Syntax

For n € w let £ n denote the propositional language of polymodal logic with a denumerable set Var = {p;|t € u;} of propositional variables and primitive symbols A, D j,..The symbols V, —>,O i,. . . On, T, J_ are defined in the usual way. A normal n-modal logic is a subset A of £n, which contains the axioms of classical logic, the formulas MD: □,(p —► q) —► (Qjp —i► □,-?) and is closed under Modus Ponens, MN: p/ü,p for t < n and Substitution. The lattice of normal n-modal logics is denoted by Afn. Given a logic A the lattice of extensions of A is £A := {0 € A/iJ0 D A }. For a modal logic A and a set of formulas T let A (T ) denote the smallest normal n-modal logic containing T. The smallest n-modal logic is denoted by Kn. The set of subformulas of a formula <f> is defined as usual and is denoted by sb(<f>). The modal depth of a formula is defined by induction: dg(p) = 0, dg{4>Aij>) = max{dg(<j>),dg(il’)}, dg(-«j>) = dg(<j>), dg{n{(f>) = dg(<f>)+\. We will often not specify the number of boxes and simply call the language £ and a corresponding n-modal logic a modal logic or simply a logic.

We associate two consequences with a modal logic: For a logic A, a set of formulas T and a formula <f> we write T 1- </> if ^ is derivable from A U T by Modus Ponens. If <f> is derivable from A U T by Modus Ponens and M N we write T l-J ^ and say that <f> is sequentially derivable from T. A formula (j> is consistent with a logic A if ~«f> £ A. A set of formulas T is consistent with A i f A $ is consistent with A for any finite subset $ of r.

For a formula <f> € £ n we will frequently use the following construction. For m 6 w the formula Om<f> is defined as follows:

□ V := 4>-□ m+V := < n).

Let a<mV := A (D' # ' < m) and define for a set of formulas T

□ (m)r := {□ * '# ' <m,<f>e r } .

□ (u,)r := € w , <t> € T }

Let Om<f> := -iC]m-id> and O l"1)^ := - ! □ ( " * ) Notice that the construction of □"*</> depends on the language. If we want to indicate that □ m<f> is defined in the language C we write a™<l>. A n-modal logic A is m-transitive if the formula □ ] —*• belongsto A. A frame for this formula is m-transitive in the sense defined on page 12.

Sometimes there is need to expand the language £ to uncountably many propositional variables {p,|t € « } , k € card. The corresponding modal language is denoted by £*. All concepts defined for £ extend without any problem to £ ".

1.2 Modal Algebras

1.2 Modal Algebras

11

An n-modal algebra is an algebra A = (A, f l , - , Di, 1) such that (A , f l ,—,1) is aboolean algebra and ü t l = 1 and 0,(0 D 6) = □ ,a fl 0,6 for i < ii. The n-modal algebras A form a variety (an equational class), denoted by M A U. A valuation of A is a map ß :Var — ► A. Any valuation can be extended to a homomorphism ß : Cn — ► A , mostly denoted by ß as well. A model is a pair (A ,ß ).

A formula (j> is consistent in A if there is a model (A,/?) with ß(<j>) > 0, we write A,ß (= (j> > 0. A set of formulas T is consistent in an algebra A if there is a model (A ,ß ) such that /?(T) := {ß(<t>)\<t> € T} has the finite intersection property, i.e. any finite intersection of elements of ß(T ) is not 0. We write (A ,ß ) (= T > 0. T is called consistent in a class K of algebras if there is an A € K such that T is consistent in A. A formula 0 is true in a model (A,/?), in symbols A,ß (= <t>, if /?(<£) = 1. A formula 0 is a theorem of A, in symbols A (= 0, if 0 is true in any model on A. The set of theorems of an algebra A is denoted by T/i(A). The theory of a class K of algebras, defined by

T ft(K ) := e K },

is a normal modal logic. Dually, for a modal logic A the class of A-algebras

V(A) := M M (=A }

is a variety. For a class K of modal algebras we denote the class V (Th {K ) ) by V (K ). This defines a Galois-correspondence between the lattice ATn and the lattice of subvarieties of M A n. For k € card and A a logic let FVa (/c) denote the «-generated free algebra in V (A ). For A € V (A ) any valuation ß : CK — ► A can be lifted to a homomorphism 7 : F t\ {k) — ► A. We will therefore sometimes denote such homomorphisms as valuations and write Fr^(p0, .. .,pK) instead of FYa ( « ) .

The concepts of subalgebras, homomorphisms, products and ultraproducts are as in universal algebra. For K a class of algebras H K (5 K , P K , /fyK) denotes the class homomorphic images (subalgebras, products, ultraproducts) of K . Given an n-modal algebra A, m £ u and a € A , Oma € A is defined analogously to Omp, p € Var. An open filter of a modal algebra A is a subset F of A which is a boolean filter of the boolean reduct of A and is closed with respect to MN: a € F =>► ü«a € F for t < n. The lattices of congruences

and open filters of a modal algebra A are isomorphic via ¥ »-► {a|(a, 1) £ ty}. The inverse homomorphism F Vp is defined by (a, 6) € if a ++ b € F. For an open filter F of A we denote A / ^ f by A /F and for a set D C A we denote the smallest open filter containing D by (D ).

A modal algebra is subdirectly irreducible, si. for short, if it has exactly one minimal non-trivial congruence. For a variety V the symbol (V/, denotes theclass of subdirectly irreducible (finite, finite and subdirectly irreducible) algebras in V.

12 1 BASIC CONCEPTS1.3 Generalized Frames

An n-frame is a structure Q — (g, (< ,jt < n ),A ) such that g is a set, (<],|i < « ) is a sequence of binary relations on g and A C 29 contains g and is closed with respect to n, — and D,a := {x 6 g\Vy £ g : x < , t/ y £ a }, i < n. The elements of A are called internal sets of Q. These form a normal n-modal algebra Q+ := (A , f l ,—,(ü,|t < n),g). Dually, given a modal algebra A, let U(A ) denote the set of ultrafilters in the boolean reduct of A. Then A+ := (f/(^4), < n),Sj.) is a frame, where

U \ < f Ui iff Va £ A : □ ,a € Ui =► a € U2 and Sa := {a + |a 6 A} with a+ = { U £ U(A)\a £ (/}.

It is easy to check that A — (A + )+ . For a cardinal k and a logic A the «-canonical A-frame is F a(k) := (F ta(k))+. The class of frames is denoted by Gfr.

A valuation of a frame Q = (g, A) is a map ß : Var — ► A. A valuation can be extended to a homomorphism ~ß : C — ► G+- Again we will often write ß instead of ß. A model is a pair (Q,ß). A formula <f> is consistent in Q if there is a model (G,ß) and x £ g with x € ß{<t>), in symbols G,ß,x f= <f>. It follows that a formula is consistent in G if and only if it is consistent in G+■ A formula <f> is true in a model (G, ß), in symbols G,ß ^ d>, if ß{4) = g- 4> is a theorem of G, G (= 4>, if <f> is true in any model {Giß)- Th{G) := {<f>\G (= <f>} is the theory of Q. For a class of frames M

Th (M ) := f){Th(G)\G € M } is the theory of M . Dually, for a logic A

Gfr{A ) := {G\G (= A }is the set of A-frames. For a class M of frames we denote the class Gfr{Th(M )) by Gfr(M). The notions of a p-morphism, a generated subframe and the disjoint union of frames are defined as usual.

There are several distinguished classes of frames. A frame (g, A) is a Kripke frame if A = 29. In this case we write g instead of {g, A). The class of Kripke frames is denoted by F t. Dually, an algebra A is a full algebra if A 2: g+ for a Kripke frame g.

A path of length m in an n-frame (g, (<l,|t < n )) is a sequence (xj|/ < m) such that for j < m there exists i < n with Xj < , * j+ i- For a C g define the m-transit of a in g by TrJ*(a) := {y € g\ there exist a path (x^jj < k) with k < m, x<j € a and x* = y}. The transit of a in g is Trg(a) := (J {? ’rj ( a)ln € w }. A frame g is m-transitive if Trg{x) C 7V£*(x) for x £ g. Call x £ g a root of g if g = Trg(x). A frame with a root is called rooted. An n-frame g is cycle free if xo # xm for any path {xj\j < m) with 0 ^ m. For an n-frame g define gx := (g ,(< i - {(y ,y )|y € <?}|t < n)) (i.e., replace all reflexive points by irreflexive points.) Then an n-frame g is T-cycle free if gx is cycle free. An n-frame (3, A) is refined if

1. Vx,y £ g : x = y & V a £ A : x £ a < $ y £ a .

2. Vx,y £ g : x < iy £ A : x £ D,o => y £ a.

The class of refined frames is denoted by Rfr. The term refined is due to T homason [72];

1.4 Completeness and Persistence 13F ine [75b] calls refined frames natural frames. A frame Q is descriptive if it is refined and f ] U ^ 0 for each U 6 U(Q+). The class of descriptive frames is denoted by Dfr. A+ is a descriptive frame for every algebra A. Given a logic A we define

Rfr(A) := Gfr{A) n Rfr,Dfr{A) := Gfr{A) Cl Dfr,F r (A ) := Gfr(A) n Fr and Fr/ (A ) := G fr (A )O F rf ,

where F rj denotes the class of finite Kripke-frames.

1.4 Completeness and Persistence

A logic A is complete if Th(Fr{A )) = A and it has the fin ite m odel property, fmp for short, if T h (F r j(A )) = A. It is easy to show that A has fmp if and only if T/i(V/(A)) = A if and only if T7i(V/.i .,.(A)) = A. Following Goranko & Passy [92] we call a logic A sequentially com plete if for all <f>, ij’ £ C:

<t> l-y[ i/> iff for all g € Fr( A ) and valuations ß : g, ß [= <f> =>• g, ß [= tp.

A logic A has sequential fmp if we have for all <j>, if) € C:

<f> iff for all g € F r j (A) and valuations ß : g,ß [= <f> =>• g,ß \= i>-

A logic A is K-compact for an infinite n € card if for T; <f> C CK the following is equivalent:

1. r i-A <t>-

2. For all g £ F r (A), x € g and valuations ß : g, ß, x |= T => g, ß, x f= <f>.

A logic is compact if it is u;-compact. A logic A is sequentially «-compact, k € card and infinite, if for all T]<j> C CK the following is equivalent:

1. r i - i * .

2. For all g € F r(h ) and valuations ß : g,ß |= T =$■<;,/?[= <f>.

A logic is sequentially compact if it is sequentially w-compact. Following G oldblatt [89] we call a logic A complex if any A-algebra is isomorphic to a subalgebra of a full A-algebra. The following definition refines this: A logic is «-complex for « € card infinite if any A-algebra of cardinality at most « is a subalgebra of a full A-algebra.

A logic A is R-persistent if g € F r (A ) whenever (g, A) € Rfr(A). F ine [75b] calls R- persistent logics natural logics. A is D-persistent if g € Fr( A ) whenever (g, A) € Dfr (A).

14 1 BASIC CONCEPTSCall a class of Kripke frames M elem entary if there is a set of sentences T in first order logic (with = , for i < n) with g a M \S g \= T. A logic A is elem entary if Fr{A) is elementary. We have the following picture of implications between the concepts introduced:

R-persistent =>• complete and elementary

D-persistent •<= complete w.r.t. an elementary class

«-compact -o sequentially «-compact o «-complex4 .

sequential fmp => sequentially-complete

fmp => complete

These implications are well-known with the exception of the line with the k's (consult Fine [75b], Goldblatt [89] and Goranko & Passy [92]). It is shown in Fine [75b] that a complete and elementary logic is not necessarily R-persistent. In WoLTER [93] is given an example of a logic with fmp but without sequential fmp. As far as I know the problems whether there are D-persistent logics which are not complete with respect to an elementary class and whether there are complete logics which are not sequentially complete are still open.The following proof makes use of the fact that a logic A is «-compact if and only if for any consistent set T with |r| < « there is a A-frame g, x € g and a valuation ß withg,ß,x t= r.

Theorem 1.4.1 Let A be a modal logic and « € card, k infinite. Then the following conditions are equivalent:

(1) A is K-compact.

(2) A is sequentially K-compact.

(3) A is K-complex.

P roo f. ( 1) => (3). Let A be an algebra in V (A ) with |.4| < « . Choose a surjective homomorphism 7 : £K — ► A. For U € U{A) the set 7~l (U ) := {^|7(<A) € U} is maximal consistent with A. Now A is K-compact and therefore for any 7- 1( {7) there is a frame gu € FV(A ) with root xu and a valuation ßu with gu,*v,ßu b 7“ 1(t f). We take the gy disjoint and define g = \J{gv\U € U (A )} and ß(pi) := \J{ßu(Pi)\U € U (A )} for * < k. It follows immediately that g € F r (A ) and that ß[CK) = {ß(<f>)\<f> € £ " } is a subalgebra of 5+ . We show that ~ß[CK\ — A. This follows if

{<!> € £«|?(^) = <7} = {4 € £ “ 17^) = 1}. But

7 (^ ) = 1 Vn 7( 0 » = 1

O VnVU € U(A) 7(Dn ) € U

1.5 Polymodal Logics 15& VnVf/ € U(A ) On<f> €O VnVtf € U(A ) 9u, ßu, xu (= □ ’>

<=> /?(</>) = 9

(3) => (2). Suppose that $ \f\ 4> with |$| < n. By «-complexity of A there is a frame (g, A) such that g is a A-frame and a valuation ß with (g, A ),ß [= $ and (g, A),ß ,x (= ->4>. Hence g,ß [= $ and g,ß,x [= -i<£.(2) => (1). Consider a A-consistent set T with |T| < « . Define $ := p; D ^ {p <j>\<f> € T } with p £ T. $ remains A-consistent: Suppose that is a finite subset of $. We may assume that = p, D(m){p —*• <f>\4> € T '} with m € w and T' a finite subset of I\ By A- consistency of T there exists a model A,ß (= T > 0 with A € V (A ). Define a valuation 7 of A by 7(p) := ß{/\ T ') and 7(q) := ß(q) for the other variables. Then A, 7 |= > 0 andis A-consistent. It follows that {p —>■ <f>\<j> € T } If sA ->p. Now, by sequential «-compactness of A, there exists a model g,ß (= {p —► 4>\<f> € T } and g,ß,x [= p. But then g,ß,x |= T. H

1.5 P o ly m o d a l Log ics

1.5.1 Fusions

Defin ition 1.5.1 An n-frame Q = (<7, (<«|i < n )) is an n-tree if it is cycle free, rooted, Oj fl <3, = 0 for i / j and each x € g has at most one predecessor with respect to the relation r := < n }. Q is a reflexive n-tree if Q is reflexive and G* is an n-tree.

Consider « monomodal logics A,-, i < n, and suppose that A,- is formulated in the language with □, for i < n. Then the fusion of (A,|t < n), <8>(A,-|i < n), is the smallest n-modal logic containing U-fA«-|i < n }. We note the following theorem from K racht & WoLTER [91].

Theorem 1.5.2 For consistent monomodal logics A,-, i < » , the following conditions arc equivalent:

1. All A,- are complete (have the fmp).

2. (g)(A,|i < n) is complete (has the fmp).

If all A, are complete with respect to finite trees (finite reflexive trees), then <g>(A,|i < n) is complete with respect to finite n-trees ( finite reflexive n-trees).

It follows that Kn (® ” T ) is complete with respect to the set of finite n-trees (finite reflexive n-trees). We will use this observation in Chapter 3.

16 1 BASIC CONCEPTS1.5.2 W h at is the upper part o f J\T2?

One prominent way to investigate the lattice of modal logics is to investigate its upper part, namely the extensions of a strong logic. In this section we show that a lattice £ A, A a bimodal logic, can be as complicated as £T even if the monomodal fragments of A are very simple. We denote the box of the monomodal language C\ by □ and define a map F : C\ — ► £ 2, <£ •-► 4>F > inductively by

PF = °2 P (<j> A iJ’)F = (f>F A if’F

(-«f>)F = O 2~'F ( in<f>)F = □ 2o 1d 2 f

For f C £ ] define VF := {<j>F\<j> € T }. Define Ai := Th( |* |) and T := A'(Op —► p).

Theorem 1.5.3 Ft : £T — ► £ (A i $ S5),T (T ) *-»• (A i (x).55)(rF), is well defined and is an embedding of £T into £ (A j M .55). Ft reflects decidability, completeness, compactness and fmp.

For the proof of this theorem we need a simulation of T -frames as (Ai(*o.55)-frames and vice versa. The idea of the simulation is simple: For a T-frame {g ,< ) simulate the points of g as equivalence-classes of a < 2-relation. These classes have to be big enough to simulate the <J-relations as a <11-relation between points in different <3 2-equivalence-classes. The exact definition is as follows: Consider a T-frame (g, < ). For x € g take a new point yx for any y with x <3 y and x y. We assume that yx ^ yz for 2 ^ x and yx ^ zx if y ^ 2 . Now define

F (x ) := { x } U {yx\x <3 y,x ^ y }, for 1 € g

F (a ) := (J {F (x )| x € a }, for a C g

< 1 := {(!/*, v)\x < y ) u { ( * , x)|i e F (g )}

< 2 := |J{F(x) x T(x)|x € p}(F (p ), < 1, <J2) is a (A i 0 55)-frame. The <3 2-equivalence-classes are exactly the sets F(x ), x € g- For G = (g ,< , A) € Gfr(T) define F [A ] := {F (a)|a € A} and

m := ™ , < i , < 2,[F [A ]]),where [F[A]] denotes the modal algebra generated by F[A], i.e. the smallest subset of F(g) containing F[A ] and closed under intersection, complement and D j,D2. The following lemma states that a union of <3 2-equivalence-classes is an internal set in F[Q] if and only if it is equal to F (a ) for an internal set a of Q.

Lem m a 1.5.4 For a T-frame Q and a C g : F (o ) e [F [A]] if and only if F (a) € F [A].

P roo f. The direction from right to left is trivial. For the other direction suppose that a C g and F (a ) € [F[A]]. Then there is a formula if) € £ 2 and o i , . . . , o n € A with

1.5 Polymodal Logics 17F(a) = rj)[F (a j),...,F (an)]. We abreviate y>[F(ai),• • •>F (an)] by V>[F(a)]. Now is closed with respect to intersection, complement and ü2 and therefore the claim of the Lemma follows if for V’ € C\:

(I) O2y>[F(a)] 6 F[A].

Claim 1 For any ij’ € C\ there exists a Ai-equivalent formula <j> with dg(<f>) < 1 and

var(<j>) = uar(V’)-

The proof of Claim 1 is by induction on the subformulas of V’- We only note the main equivalences in A\. They are easily checked on | • -* [.

Aj I- ODp *-* (- ’p A Op) V Op

Aj I- OOp *■+ Op and Aj b Oüp <-► DOp

Aj I- 0 (0 p A q) «-*• (Op A q) V 0 (p A q)

Aj b 0 (-i0p A q) <-*■ (~>Op A q) V (p A 0(->p A q)) V (-»q A O(~>p A q)) H

We need some notation for normal forms in modal logic. Fix variables p i , . .. ,pn. Then the set of atoms in n variables is := {jripi A .. . A 7rnpn|7r, = blank or 1r,- = -•}. For all partitions (A"+ , X ~ ) of Atn and Q € Atn

P (Q ,X + ,X ~ ) := Q A € X +) A A (-^ ? il< ? i € X ~)

is a normal form of degree 1. By a result of F ine [75a] any formula <t> with dg{<f>) < 1 is equivalent to J_ or a disjunction of normal forms of degree 1. Hence, by Claim 1, (I ) reduces to

(II) Oa \/(Pi[F(a)}\i 6 /) € F[A] for {P,|r€ /} a set of nf’s of degree 1.

Now 0 2 V(Pf[P(fl)]|t € /) = V ( < 2P«[P(fl)]|i € I ) and therefore (II ) follows immdiately if 0 2P [P (a )] € F[A] for all normal forms P of degree 1. We fix a P = P(Q, X + , X~).Case 1. Q & X +. Then 0 2P [P (a )] = 0 because A i b p —► OpCase 2. |X+ 1 > 3. Then 0 2P [P (a )] = 0 because no point in a Ai-frame has more than two successors.Case 3. X+ = {(?}. Then 02P[P(a)] = P(g[0l,.. .,an]) € F[A].Case 4. X + = {Q ,Q i} with Q Q\.Then 0 2P [P (a )] = F(Q A O t ^ d , . . . , cn]) € F[A]. H

Now consider a (A i $55)-frame (h, < 1, <J2). The <J 2-equivalence-class of an element x € h is denoted by [*]. Define

P _ 1(c) := {[x]|x € c }, for c C h[x] <J [y] iff there are Xi e*[x], yi € [y] with Xi < !

18 1 BASIC CONCEPTS{F l (h), < ) is a T-frame. For a (A ® 55)-frame H = (h, < j , <12, B) define

F~\H ) := ( F - 1(/ i),< l,F -1[£]>, where F -1 \B) := { F _1(6)|&€ £ }• Note that F _ 1(fc) = F -1 (0 26) for all b C h and therefore

F -1 [P] = {F -1(6)|6 = 0 2fc,6 € B }= {F _1(6)|6 = D2b,b € B )

= { F - ' ( 6 ) |6 = U M x 6 6},6 6B }.We postpone the proof that is a frame, i.e., that F~l [B] is closed with respect tointersection, complement and □, and first prove the following more general lemma.

Lemma 1 .5.5 Suppose that ß is a valuation on (F a(/i),<j) and 7 is a valuation on h with i ( a 2p) = LK M IW € ß(p)} for p € Var. Then 7(<t>F) = U iW IW € ß{<f>)} for<t> € C\.

Proo f. We show that for x 6 h and <f> € C\ : [x] € ß(<j>) iff [x] C y(<f>F ). From this follows that 7(<t>F) = LKM K1] € /?($)} because 7 (<f>F) is a union of < 2-equivalence-classes. The proof is by induction on the subformulas of <f>.

[x] € ß(p) [x] € ß(<f> A ij>)

[x] € ß(-«f>)

[x] € ß(a<t>)

O

o

o

o

[1] c 7 (a 2p) = 7(pF )-

[x] C 7(< f ) and [x] C 7(V»F)

I*] C 7((V>A V0F)-[x] $ ß(<f>)[x] % 7(<f>F )[x] C 0 2- ‘y(<f>F ) = 7 ((- ’^ )F )-

V [p]: [x] < [p] =* [y] € ß(<f>)Vxj € [x]Vp e h : xi <1 x y ^ [p] C 7 (<j>F )

[x] C □ 2a 1a 27 (^F) = 7 ( ( ° ^ ) f ). H

We now prove that F - 1(W ) is a frame. It is enough to show that for a formula <f> € C\ and 6i , . . . , f c n e B with 6,- = D26i for * < » , ^ [F - 1(6j ) , . . .F - 1(6n)] € F _ 1[B]. But, by Lemma 1.5.5, <l>F[bi,.. . ,6n] = CJ{[x]|[x] € V’[F _ 1(6i ) , .. . F _ 1(6n) ] } e B and therefore ^ [F - 1(h i) ,•. .F - 1(6„ )] = F - 1(< F [6i , . . . , 6n]) € F _ 1[B].

Lemma 1.5.6 For 7i e Gfr(A\ (8) 55) and <)> € C\ : F *(W ) ^ <j> iff "H ^ <f>F. For 6 € G fr{T ): 0 M iff F(G) |= <f>F .

P ro o f. Suppose that F 1 (7f), ß, [x] |= <f>. Define a valuation 7 on H by 7(p) := LK b llb ] e ß{p)} for p eV a r.

7 is well-defined by definition of F~l (H ) and 7(ü 2p) = 7 (p). Hence, by Lemma 1.5.5, 7 (^F) = l i t t * ] € ß(<f>)}, and H, 7 ,x |= <f>F. Conversely, suppose that W ,7 ,x <j>F.

1.5 Polymodal Logics 19

Define a valuation ß on F - 1 (W ) by ß (p ) := {[p]|p € 7 ( 0 2p )}.Then 7 ( 0 2p) = (J{[p]|[p] € ß (p )} and, by Lemma 1.5.5, F -1 (W ),7 , [z] (= 4>- The first equivalence is shown. The second equivalence follows directly from F ~ lF [G } — G for G € G fr(T ). This is a consequence of Lemma 1.5.4: Let G = (g ,A ). Then

F ~ XF[G ] = < {F (* )| * € <7},<J,F-M F[yl]]).It is easy to check that x h-> F (x ), x € g, defines an isomorphism between the Kripke- frames. It is an isomorphism between the modal algebras of the internal sets because for a C j : {F (x )| x € a } € F -1 ([F [A ]] ) if and only if lJ {F (x )|x € o } € [F [A ]] if and only if (by Lemma 1.5.4) F (a ) = (J {F (x )|x € a ) € F [j4] if and only if o € A. H.

P ro o f of Theorem 1.5.3. Consider two sets Ti, r 2 C C\. By Lemma 1.5.6, ft T f if and only if F ~ l (f t ) f= I\ for ft € Gfr(A\ c*>55). It follows that (A i c * )5 5 )(r f ) = (A i <x i55 )(rf) if T (T i ) = r ( r 2). Thus, Ft is well-defined. Now suppose that 7 \ r i ) ^ T ( r 2). Let G [= Ti and G ¥ r 2. Then F (G ) T f but F (G ) ^ r£\ Hence Ft is injective.F r (V (0 t l* € / )) = V (F r (0 < )l* € I ) is clear.Hence Ft is an embedding if Ft ( 0 i ) fl Ft ( 0 2) = Ft ( 0 i n 0 2) for 0 i , 0 2 € ST. This follows if Ft ( 0 i ) D Ft ( 0 2) is equal to (A j $ 5 5 ) (r F ) for a set T C C\. Assume that Ft ( 0 i ) = (A i o o 5 5 )(r f ) and Ft ( 0 2) = (A i $ S 5 )(r . f ) with v o r (r i )n va r(T 2) = 0. Then

we have for A := Ai $ 55

Ft ( 0 O n Ft ( 0 2) = A ({a £ y > F v o^ ij,F\4, e r u j e r 2, m e « } )

= A ( { (D 2D i D2)to f V ( □ 2D1D2)my>F |</> € Tu € r2, m 6 w } )

= A ( { (G £ j^ V V’) F|<A € T j, V’ € r2, m € w } )

It is easy to check that Ft reflects decidability. Suppose that Ft ( A ) is complete. Then A is complete with respect to jF_1(/i), h £ F r (F j (A )): Suppose that <f> £ A. Then </>F £ Fp (A ). By completeness there exists h £ F r (F r {A ) ) with h ^ </>F . But then, by Lemma 1.5.6, F~l (h) ft <f>. That Ft reflects fmp and compactness can be proved analogously.-!

A few remarks on this embedding. It is easy to check that Ft is not only an embedding of ST in £ (A i (X) 55), but an embedding into the interval [Ai # 55, (A i $ 5 5 ) (d 2p —► D ip)]. It follows that the images are conservative extensions of the monomodal fragments Ai and 55. The simulation of monomodal frames as bimodal frames by giving the points (or possible worlds) the structure of 55-frames seems to be quite natural - not only from a mathematical point of view. There are obvious similarities to the constructions of modal predicate logic and to the embeddings of the lattice of intermediate logics into 54, which might be of interest for future investigation.

Let us note the main examples of monomodal logics A i, A 2 such that £ (A i® A 2) behaves better. For n £ w ,n ^ 0 define Altn := f\(Opi\i < n + 1 ) -+ V (^ (j*iA p j)| t ^ j ; i j < n + 1 ). The proof of the following theorem is a simple extension of the monomodal proof in Bellissima [88].

T h eo rem 1.5.7 For all m ,n > 0 any extension of <$nK.Altm is R-persistent. Especially for any two tabular logics A i, A 2 any extension of Ai ® A 2 ts R-persistent.-\

20 2 SUBLATTICES OF AfN2 Sublattices of Afn

For a logic A the map A : T »-► A(r),T C £, is an algebraic closure operator on £. The closed elements with respect to this operator are the normal modal logics above A. Thus the normal modal logics above A form a complete lattice, £ A. (Note that we use the same symbol A for a logic and the corresponding closure operator.)In this chapter we will study certain kinds of sublattices of £A. The following easy and well-known lattice-theoretic concept is fundamental to our investigation. We assume that a complete sublattice V of a complete lattice T contains the maximal and minimal element of T.

Definition 2.0.8 Suppose that V is a complete sublattice of a complete lattice T . Then for a € T the upward (downward) projection a fp (alz>) of a is defined by

a]v = /\(b € V\b > a)

aiv = \/(beV\b<a)

Now consider a complete sublattice V of £ A. Then we call V a compactness preserving sublattice of £ A, a c.p. sublattice of £A for short, if there exists a map /p : C — ► £, <t> <h> such that A(<£)Tp= A(^p) for ^ 6 C. Hence, by some lattice-theoretic manipulations, V is a c.p. sublattice of £A if and only if for any logic 0 6 fA the logic © |p is compact in £A if © is compact in £A. This definition is not very informative, if one tries to find (interesting) c.p. sublattices of the lattice of extensions of a logic. Before giving an equivalent notion, which will be more useful, we note the following charactarization of the finite c.p. sublattices of the lattice of extensions of a logic.

Proposition 2.0.9 A finite sublattice V of £A is a c.p. sublattice of £A if and only if it contains C and A and any 0 € P is finitely axiomatizable above A.

2.1 Describable Operations

Consider a variety V of modal algebras and suppose that C : V —► {K|K C V } is an operation on V. Define C (K ) := \J{C(A)\A € K } for K C V. We call C a describable operation on V if

C I C C (K ) = C (K ) and K C C (K ) for K C V.

C II There is a map f c : C — ► C, <!>*-+ 0C, such that for any <j> € C and A € V : A (= <j>c iff C{A) [= <t>.

<j>c is called a C-formula of <f>. If C is a describable operation and there are two different maps fCl and with property (C II), then the formulas 4>Cl and are deductively

2.1 Describable Operations 21

equivalent above A for all <t> € C. (We call two formulas <t>u<fa deductively equivalent above A if A (0 i) = A(<fo)0 Suppose not: Then there is an algebra B € V with B |= <f>Cl and B ^ <t>°2- But then C (B ) f= <t> and C (B ) ^ <£, which is a contradiction. We assume that for a given describable operation C the the map fc is fixed. Any formula deductively equivalent to a C-formula with respect to this fixed fc will be called a C-formula too.

Definition 2.1.1 Let C be a describable-operation on V. A variety W C V is called a C-variety if C (W ) = W . Dually, a logic A is a C-logic if V (A ) is a C-variety.

Simple examples of describable operations on V are the operations Identity,/,# and S. In these cases (C I) is clearly satisfied and (C II) works with <j)c = <j>. Therefore anything we prove for all C -varieties or C-logics holds for all modal varieties respectively logics with C = Identity. It should be clear how to define the notion of a describable operation on generalized frames. Hence we will call an operation on Gfr which satisfies the conditions (C I),(C II) a describable operation as well.

Proposition 2.1.2 Suppose that C fulfills condition (C I ) for describable operations. Then (1) and (2) are equivalent:

(1) C is a describable operation via fc : 4> <f>c.

(2) There is a map fd : £ — ► <f>d} such that for all B € V : <f>d is consistent in B if and only if cj> is consistent in C (B ).

For a given map fc with property (1) the formula <j>d := " ,(("»<^)c) satisfies (2) and for a given map fd with property (2) the formula <ff := -»((-»<£)**) satisfies (1). <f>d is called a dual C-form ula of <j>.

Proof. We show the direction from (1) to (2). Define <j>d = -»((-»0 )c) and let B € V. Then <j>d is consistent in B iff B ^ (~'<t>)c iff C {B ) ^ -»<£ iff <j> is consistent in C{B ). H

The following proposition lists some easily proved but useful properties of describable operations.

Proposition 2.1.3 Let C be a describable operation on V and A = Th(V). Then we have for all formulas <j>:(0 A(4) C A(0C) (2) A(0C) = A(<n (S) A(0) = A(V0 => A(0C) = A(^e).

Proposition 2.1.4 For a C-logic A and 0 € LA the following conditions are equivalent:

(1) 0 is a C-logic.

(2) 7/0 = A(0 , then A(rc) = A(r).

22 2 SUBLATTICES OF AfN(S) © is complete with respect to a class K with C (K ) C K .

P roo f. The direction from (1) to (3) is trivial. Now suppose that (3) holds with respect to K and that 0 = A(r). A (T ) C A (TC) follows from Prop. 2.1.3. Suppose that <f> € T. Then <f>c € A (T ) if and only if we have for every A € K that A (= <f>c. For A € K we have A |= <t> and it follows C{A) [= <f> because C{A) C K . This implies A \= <pc and rc C A(r) follows. It remains to show that ( 1) follows from (2). Consider A € V (0 ) and take a set T with 0 = A(r). Then A |= Tc and therefore C(A) |= T. This implies C{A) C V (0 ).

It follows from Prop. 2.1.4 that a logic 0 £ £A is a C-logic iff 0 is axiomatizable by C-formulas iff for all </> € C : <f> € 0 implies <f>c £ 0.

Theorem 2.1.5 For a logic A and a set of logics D C £A the following conditions arc equivalent:

(1) D is a c.p. sublatticc of € A.

(2) There is a describable operation C on V (A ) such that the C-logics are precisely the logics in D.

If C satisfies (2) and f c : <f> <f>c has property (C I I ) then A {<f>c) = A(4>) to for <f> € £.

P roo f. Suppose that D is a compactness preserving sublattice of £A. Now define C (A ) := V(Th(A ) |r») and 4>c := 4>d for <f> € C. Suppose that K C V (A ). Then from A £ K follows A £ C (K ) because A f= T h (A )io - If A € C C (K ) then A |= Th(B )|o for a B € C '(K ). But for this B there exists a C € K such that B |= Th(C )[o- Now Th{A) l D D Th(B) |o D Th{C) |D and therefore A € C (K ). For A € V (A ) and ^ £the following equivalences hold: 4 iff Th(A ) D A{4>d ) iff Th(A ) |r> 2 A (4>d ) iffTh(A ) I d 2 A (<t>) iff C (A ) |= <f>. It is shown that C is a describable operation on V (A). Now suppose that 0 £ f.A. 0 = A(r) for a set T of formulas. But then 0 € D iff A({<£d |> € T }) = 0 iff (by Prop. 2.1.4) 0 is a C-logic.Now suppose that C is a describable operation on V = V (A ). We show that the C-logics form a compactness preserving sublattice of £A. A is a C-logic because C (V ) C V and £ is a C-logic because C (0) C 0. Consider a set of C-logics {©«|t 6 / }. Then, by Prop.2.1.4, there exist sets T, with 0,- = A(T?) for * € /. But then, by Prop. 2.1.4,

vm € /) = V(A(r?)|t € /) = A(U{r?|t € /})is a C-logic. I f <f> € PK®«!* € / }, then <f>e € PK®«!* € / }, by Prop. 2.1.4. Again, by Prop.2.1.4, it follows that P){0i|* € / } is a C-logic. We have shown that the C-logics form ?. complete sublattice V of £A. Consider a logic A c l e a r l y A (<f>) C A(<f>c) and, by Prop.2.1.4, any C-logic which contains <f> also contains <f>c. Thus, A (^ )| p = A (^c). H

From now on we will use the notation of describable operations for compactness preserving sublattices of the lattice of extensions of a logic. For a describable operation C on

2.2 Subframe Logics 23V (A ) the lattice of C-logics above A is denoted by CA. The upward and downward projections of a logic 0 £ £A are denoted by 0 j c and 0 | c. The projections are represented by 0 | c= \{{<t>c\4> £ 0 } ) and 0 | c= A({<f>c\<f>c £ 0 } ) .

Corollary 2.1.6 Suppose that C is a describable operation on V (A ). Then the map Ac : >-* A(rc) n Cc, r C £, is an algebraic closure operator on Cc such that the latticeof closed elements with respect to Ac is isomorphic to CA via F : 0 0 n Cc.

Definition 2.1.7 A describable operation C on V is a good operation, if

(1) Cc is decidable and f c : <t> >-> <t>c is computable.

(2) If A £ V is finite then C (A ) is a finite set of finite algebras.

(3) The class of full algebras is closed under C .

If a describable operation on V (A ) is good, then nearly all interesting questions concerning logics in CA can be translated into questions about Ac and Cc :

Theorem 2.1.8 Let C be a good operation and 0 a C-logic. Then the following holds:

(1) 0 is decidable if and only if 0 Cl Cc is decidable.

(2) 0 has fmp if and only if for any <t>c 0 there is a finite Q-algebra A with A\fc 4>c.

(3) 0 is complete if and only if for any <f>c £ 0 there is a 0 -frame g with g <frc.

Proo f. ( 1) The decidability of 0 n Cc follows from the decidability of 0 and Cc. Now suppose that 0 n Cc is decidable and take <f> £ C. Then compute <f>c. Now <f> £ 0 if and only if <f>c £ 0 D £ c, which is decidable.(2) The direction from left to right is trivial. For the other direction take <f> £ Q. Then <t>c & 0 and there exists a finite 0 -algebra A with A ^ <f>c. But then C(A) <f>.(3) is proved analogously.-!

We now introduce some examples of describable operations.

2.2 Su bfram e Log ics

For frames Q = (g, (< «jt < n), A) and Tt we call Tt a subframe of Q induced by h £ A if H — Gh •’= (h, (<,• n h x h\i < n), A n h),

where A Cl h denotes {a n /i|a £ A}. In this case we write 7i C Q. Given a frame Q and h £ A, the structure Gh is always a subframe of G- For the proof it is enough to show that

24 2 SUBLATTICES OF AT»A n h is closed with respect to intersection, complement (relative to h) and

□ f (a n h) := { x G h|Vp € h : x < , y ^ y € a 0 h) .We show closure with respect to □*.

For a € A : ü^(a H h) {x G h|Vp € h. : x y £ a n h } □,(/i -*• a n h) fl h G A fl h.

We remark that a generated subframe (/i, B) of (g, A) is not a subframe of (g, A) in general, namely if h £ A.We define an operation Sf on Gfr by Sf(G) = {'H\H C (?}. Notice that Sf has the following closure properties: (I ) Sf{Rfr) C Rfr and (I I ) Sf(Dfr) C Dfr. (I ) follows immediately from the definition. For (II ) suppose that Q is descriptive, that h G G+ and that U is an ultrafilter in G* ■ Then U C G+ has the finite intersection property and therefore f ) U ^ 0. r \ v n h ? 9 follows immediately.

Following the definition of a subframe we now define the notion of a subframe algebra. For modal algebras A and B and 6 G A the algebra B is a subframe algebra of A induced by b, in symbols B C A, if B Ab := (A 0 6, n6, —b, (Oj|* < n), 0,6), whereAC\b = { a 0 6|aG>l}(aj n b) n6 (a2 nb) = a\ n 02 n b—6(a n b) = - a n b□ J (a n 6) = ü,(ö —► a) n 6, i < n.

The boolean reducts of the subframe algebras of a modal algebra A are obviously the relativizations of the boolean reduct of A. Again for each b G A the algebra Ab is a subframe algebra of A. This can be shown directly but also follows, by representation of modal algebras as generalized frames, from the following easily proved facts:

For all G and h £ Q+ : (& )+ = (G+ )h.

For all A and a G A : (A 0)+ (A + )a+.

Define Sf(A) = {B\B Q A}. Sf satisfies condition (C I) for describable operations: K C Sf(K ) is clear. For Sf(Sf (K ) ) = 5/ (K ) suppose that b G A and a G Ab. Then a e A and a < b. Now it follows by definition that A a = (A t )0. Hence Sf is a describable operation on M A n for n G u>, if for any <f> G C there exists a formula 4>s* such that for B G M A n : B |= 4>S} iff Sf(B) (= <f>.For a formula <f> we define <f> | p inductively via q lp = ql\p(^ A V>) iP = ( ^ i p ) A ( V » i p )(-><(>) i p = I p) A p(□«•^) |p = □.(p <f> I p) A p, * < n.

The following lemma is the algebraic translation of a lemma proved in K racht [90b].

2.2 Subframe Logics 25

L em m a 2 .2 .1 Suppose that b £ A , ß : Var —* A and : Var Ab are such that ß (p ) = b and 7 (q ) = ß (q ) fl b for all q £ Var, then ß(<j) \ p) =

We are now ready to prove that 5/ is a describable operation.

T h eo rem 2 .2 .2 Forn 6 u the operation Sf is a good describable operation on M A U with Sf -formula <j)Sf = p -> <t> l p and dual Sf-formula <f> ] p for <f> £ C arid p £ var{(j>).

P ro o f. We first show that <f> j p is a dual Sf - formula. Suppose that <f> j p is consistent in B. Then there is a valuation ß of B with ß{<f> l p ) > 0. It is easy to show that <f> j p has the form Ap for a suitable V>. Hence /j(p) > 0. Now define a valuation 7 of #£ (p) by 7 (g ) = /?(p) n /?(<7). By Lemma 2.2.1, 7 (<£) = ß(<f> 1 p ) > 0 and therefore <f> is consistent in

B. Conversely, suppose that there is a b £ B and a valuation 7 of with 7 (<£) > 0. We

define a valuation ß of B by /?(p) := b and ß (q ) := 7 ( 9) for q ^ p. It follows from Lemma2.2.1 that ß(<f> 1 p) = 7 (<f>) > 0; here we need that p £ var{<f>). It remains to show that P —► <t> I P is a ^/-formula of <f>. But —»((—«0) j p ) = I p ) A p) = p —► \ p, andtherefore p —► <f> l p is a 5/-formula, by Lemma 2 .1 .2 . It is now trivial that Sf is a good

operation. H

A logic A is a subframe logic if Gfr{A ) is closed under subframes, or, equivalently, if V (A ) is closed under subframe algebras. Some remarks concerning subframe logics. It follows

from Prop. 2.1.4 that a complete logic A is a subframe logic if and only if F r {A ) is closed under subframes. The subframes of a Kripke frame g coincide with the substructures of g. Subframe logics above K4 were introduced in F in e [85] in a different way. We will show

in Section 3.1.2 that for logics above K4 both definitions are equivalent.

2 .2 .1 E xam p les

The following formulas <t> are 5/-formulas because respectively # 2(^)5 is completewith* respect to a class of frames closed with respect to subframes:

1 . Dp, Dp p, Dp —► p, Dp —► □ □ p, Op —► OOp.

2. D (O p - t p ) - * Op, 0 (0 (p - » Op) - » p ) — p).

3. B\ := p —► OOp.

4. For n € <•>,«» # 0 : /„ := A(<>P.|* < « + 1 } -*■ V (0 (P .A (p jV O p i ))|i ^ j ; ; i , j < n + 1 ).

5. For n £ u;,n / 0 : Altn.

6. t := {p —► Ü 2<>ip;p —► üi<>2p }.

7. Ui := Ü 2P —* EiP» U2 := D ip —► Ü 2P and Id = { f/i, f/2} .

26 2 SUBLATTICES OF Mn

Now it is easy to check that the following logics are subframe logics:

1. Kn for n € u , Ver = K(Dp),Triv = K(Op ++ p), K4 = K(Dp —► GGp),T = K(Op -H. p), 54 = K4.T, Kb = K(Op -* DOp), K.B\,G = ^ 4(D(Dp -+ p) -*• Op), Grz = 54(D(D(p Op) - p) — p)).

2. For any subframe logic A the logics A .Altn and A./n are subframe logics.

3. For any two monomodal subframe logics Ai and A2 the logic Aj c-0 A2 is a subframe logic.

4. For a bimodal subframe logic A the logics A.t, A.U\ and A.Id are subframe logics.

All the logics defined are well-known from the literature (consult SEGERBERG [71] or Rauten berg [79]) with the exception of and K-z.Id. It is easy to show that K-i-U\is the theory of the class of 2-frames (<7, ,n ^ 0, define Trn := (G ^ p —► D ^ p )^ . By definition, the logics K.Trn are subframe logics.It is easy to show that (O n+1p —*■ O(n)p)s is a Sahlqvist-formula. (For a definition of Sahlqvist-formulas consult Sahlqvist [75], Sambin & Vaccaro [89] or Kracht [91a].) Hence K.Trn is elementary and complete. A frame g is a K.Trn-frames if and only if any finite subframe of g is n-transitive if and only if for any path (*,•(* < n + 1) there exists a subpath (y,-|i < m) with m < n and yo = xq and ym = xn- It is worthwile to note that from completeness and elementarily of a logic A does not necessarily follow that A is elementary. As an example take A := A '4 (G± V OGJ_). This logic is elementary and complete, but G = A (this will be shown on page 70) and G is not elementary.

2.3 C on fin a l Su b fram e L og ics and o th e r exam p les

One possiblity to define more describable operations is to restrict the forming of subframes to sets which are definable by a single formula. Consider a formula 4>(p\,.. ,,pn), a frame H = {h, (<i|t < n),A) and g € A. Then G — Hg is a ^-subframe of H if there exist 61, . . . , bn € A with <f>[b\,...,bn] = g. We write G H. Clearly a subframe is simply a p-subframe. For <f> € C define an operation Sfj, on Gfr by Sf^(H) := {G\G Q<f. H ). For a formula and p € Var with var(ij>) 0 var(<f>) = 0 and p var(if>) let if) J, <j> := if) j p[4>/p]. It is a straightforward extension of the proof of Theorem 2.2.2 to show

Lem m a 2.3.1 For all 4>, i/> with var(<f>) 0 var(if>) = 0 and H € Gfr:

Hence 5/ is a describable operation on Gfr(A ) with describing formula — <j> \ <j>if SU(Gfr(A )) C Gfr(A) and

2.4 Splittings in Modal Logic 27

( 1 ) YH. = (h, A) e Gfr(A) 36 6 A : # ] = h.

(2 ) VW € G/r(A ) V6 £ A Vc £ {a 0 <f>[b]\a € A } 3d € A with (W ^)<*[c ] =

An interesting example is the operation 5/con with Con := p A d (p V Op).

T h eo rem 2.3.2 Sfcon is a good operation on G fr (K 4).

P ro o f. We only have to check the conditions (1 ) and (2). Let 7t = (/i, < , A ) € (7/r(A'4).(1 ) follows from Con[h\ = /i. Call a set c C h confinal in W if for all x € c and y £ T rh{x ) we have T r^ y ) He / 0. It is straightforward to prove that a set d € A is confinal in H if and only if there exists a set 6 E A with d = Con[b]. (2 ) follows immediately from this

observation. H

It follows that the Sfcon-logics above K 4 are precisely the confinal subframe logics defined

in Zakharyaschev [92].

2.4 S p lit t in g s in M o d a l L o g ic

2.4.1 Splittings

Suppose that V = (Z ) ,V ,A ,0 ,1) is a complete lattice. An element po G D splits P , if there is a p\ £ V such that for any a € D : a < po or a > p\ but not both. In this case pi is uniquely determined by po and we say that pi is the sp litting -com pan ion of po. Pi is denoted by P/po- (po»Pi) is called a sp litting -pa ir. If pi is a splitting-companion we say that p\ co-splits P or simply that pi is a sp litting of P .For a € P let 5a := {b € P |6 > a } denote the lattice of extensions of a. If po € £a splits 5a we simply denote the splitting companion of p0 by a/p0 and call it a splitting of a. An

element a £ D is (strongly) irreduc ib le if for ail sequences (6t*|t £ I ) with a = A (M l € I ) there is an * £ I with a = 6t. An element a is (strongly) p r im e if for a > A(&*ll € /) there is an i £ / with a > 6t.

P rop os it ion 2.4.1 a £ D is irreducible in V if a is prime in P .

P ro p o s it io n 2.4.2 a £ D is prime in V iff a splits P .

Hence splittings can be visualized as follows:

Splittings can be iterated. An element pi € D is a union-splitting of V by F C V if all Po € F split V and p\ = V F. p\ is denoted by V/F. Again for an element a € D and F C £a we call a union-splitting £a/F a union-splitting of a and denote it by a/F.

Proposition 2.4.3 Suppose that p\ = V/F. Then we have for all a € D : a > p\ if and only if a £ p for all p € F.

An element p\ 6 V is an iterated splitting of V by (F\,... Fn), Fi C D, if any Po € F\ splits V and for 1 < t < n — 1 any po € F;+i splits T>/Fi/F2/ .. ./F, and pi = V/F1/F2/ .../F n_i/Fn. An iterated splitting pi = X>/Fi /F2/ .. ./Fn_i/F„ is a fin ite iterated splitting if Fi U . . . U Fn is finite.

Proposition 2.4.4 Suppose that p\ = U/F1/F2 .. ./Fn. Then we have for all a € D : a > pi if and only if a £ p for all p € Fi U F2 . . . U Fn.

The following proposition gives some information about the connection of splittings in different lattices.

Proposition 2.4.5 Suppose that T is a complete sublattice of V and that (po,pi) is a splitting pair in V. Then (pol^ jP iT^) »* a splitting-pair in T .

P roo f. We write p | instead of pt^"- Let a € F . Then a < po or a > p\ and not both. Case 1. a < po- Then a < poi, because a € and a ^ pi because a ^ pi and pi < pi |. Case 2. a > pj. Again, since a € F , we know that a > pi |. a ^ po 1 follows from a ^ po and po > Pol • H

We have the following picture:

2.4.2 A General Splitting-Theorem


In this chapter we give an algebraic and a syntactic characterization of splittings of compactness preserving sublattices of £A. The exposition is inspired by Kracht [90a], who investigates splittings of £A. The main differences apart from the one mentioned above are that we do not restrict attention to finitely presentable algebras and that we do not use but prove Jönssons Lemma. In this chapter C denotes a describable operation on V (A ) and 0 a logic in C A. Since any logic which splits C A is irreducible in C A let us first prove a necessary condition for irreducibility of a logic.

Proposition 2 .4.6 A logic 0 is irreducible in C A only if£> = T h (C (A )) for a finitely generated subdirectly irreducible, algebra A.

Proof. Suppose that 0 is irreducible in C A. As is well-known 0 = € /}with Ai subdirectly irreducible and finitely generated for i £ I. Now 0 is a C-logic and therefore 0 = f| {T / i(C (A ))h € /}. By irreducibility of 0 in C A and T h (C (A i)) £ CA for all i £ I it follows that 0 = Th (C (A i)) for an i £ I. H

Subdirectly irreducible modal algebras are well-understood by the following proposition proved in Rauten berg [79].

Proposition 2.4.7 (Rautenberg) A modal algebra A is subdirectly irreducible if and only if there is an o £ A with the following property: o / 1 and for all a £ A , if a ^ 1 , then there is an m £ lj with o > G(m)a. Such an element o is called an opremum of A. A 7i opremum has the property that for an open filter F either F = {1 } or o £ F.

Consider a Kripke-frame (g, < ) with root x. It is straightforward to check that g — {x } is an opremum of g+. Hence g+ is a subdirectly irreducible algebra. It is readily shown that a finite frame g is rooted if and only if $r+ is subdirectly irreducible.

Lemma 2.4.8 Let T C Ck,k £ card, be a set of formulas and let K be a class of modal algebras. Then T is consistent in soine C £ Pu (K ) if and only if any finite subset ofT is consistent in K .

Proof. A simple application of classical model theory. H

Proposition 2.4.9 A set of formulas G ^ r ;0 C £ K is consistent in an algebra B if and only if there is an A £ H(B ) and a valuation ß of A with A ,ß (= T and A,ß\=<l>> 0.

Proof. Take a valuation 7 of B such that <j>) = { 7 (V;)IV> € G ^ r ; ^ } has thefinite intersection property. Let F be the open filter generated by 7 (G ^^ r ) in B. Now define A := B /F and let pr : B — ► A be the canonical homomorphism, ß := pr o 7 is a valuation of A with A,ß (= T and A>ß |= <t> > 0. The converse direction is trivial.H

Definition 2.4.10 Suppose that A € V».,-.(A) is generated by {o ,a i,.. .,aK}, k € card, and leto be an opremum of A. Define s : Fr\{p0,p\,.. .,pK) — ► A by p0 o and p, a,. Then s is surjective. Now define F := {</>!$(</>) = 1 } and let pr : Fr\(p) — ► Fr\(p)/F be the canonical homomorphism. Let i : Fr^(p)/F — ► A be the uniquely determined isomorphism such that i o pr = s. Any A C CK with (A ) = F is called a diagram for A with respect to A. We call A finitely presentable above A if A has a finite diagram with respect to A.

Proposition 2.4.11 Suppose that A € VS.,(A), that A is a diagram of A with respect to A and that B € V (A ). Then D ^ A ; ->pD is consistent in B if and only if A E HS(B).

Proo f. Suppose that □ ^ A j- 'P o is consistent in B. By Prop. 2.4.9 there exist C E H(B) and ß : Fr\(p) — ► C with /3(A) = 1 and ß{~>p0) > 0. From (A ) C {t/>\ß(r/’) = 1} follows the existence of / : Fr\(p)/(A ) — ► C with / o pr = ß. We show that / is injective. pr(p0) is an opremum of F rA(p )/ (A ) and thus injectivity follows if f(pr(p0)) ^ 1. But this follows from ß(p0) if 1. We have shown that F rA(p )/ (A ) € SH(B) and it follows that A € SH(B). But then A is in HS(B).Conversely, suppose that A € HS(B). By definition, D ^ A ; ->p0 is consistent in A. Now it is standard to check that A ; ~>p0 is consistent in B. H

Theorem 2.4.12 Let C be a describable operation on V and let A € V,.,-. Suppose that A is a diagram of A with respect to Th(V) and that K C V. Then the following conditions are equivalent:

(1) A € HSPVC{K ).

(2) A € V (C (K )).

(3) Any finite subset of D ^ A ; ->p0 is consistent in C (K ).

(4) Any finite subset of (O ^ A ; ~'P0)d is consistent in K .

P roo f. ( 1) ^ (2) and (2) =» (3) can be proved by standard methods. (3) (4) followsfrom the definition of dual C-formulas. We show that ( 1) follows from (3). Suppose that any finite subset of A ; ~>p0 is consistent in C (K ). Then, by Lemma 2.4.8, there is a C £ Pjj(C (K ) ) with q M A ; ->p0 consistent in C. Hence, by Prop. 2.4.11, A € HS{C). H

Notice that for C = Identity we have proved Jonssons Lemma for any variety of modal algebras. In case of m-transitive varieties and finite diagrams we do not need ultraproducts.

C orollary 2.4.13 Suppose that C is a describable operation on V, A = Th(V) is m- transitive and that A E Va.t. has a finite diagram A with respect to A, i.e. A is finitely presentable. Then the following conditions are equivalent for K C V;

312.4 Splittings in Modal Logic

( 1) A € HSC{K ).

(S) A e V(C(K )).

(3) □<”*)A; -'Po is consistent in C (K ).

(4) ( ° (m,A A - Po)d is consistent in K .

Proof. The only implication which does not directly follow from Theorem 2.4.12 is (3) =}► (1). □ (n‘)A ;-«p0 is consistent in C (K ) if and only if □ (“') A ; ->p0 is consistent in C (K ) if and only if A € HSC{K ), by Prop. 2.4.11. H

We now come to the main theorem of this chapter. Let us first define some notations for splittings. It is said that a logic 0 C-splits A if it splits CA. The splitting-companion of 0 is denoted by A/c0 and we call it a C-splitting of A. By Prop. 2.4.6, a logic 0 which splits CA is representable as 0 = Th{CA) for a subdirectly irreducible finitely generated algebra A. Therefore we say that A C-splits A if Th(CA) splits CA. A /cTh (C (A )) is denoted by A/CA.

Theorem 2.4.14 Suppose that 0 = Th(C (A )) with A € Va.,(A) finitely generated, A a C-logic. Let A be a diagram of A with respect to A. Then the following conditions are equivalent:

( 1) 0 splits CA.

(2) There are m € u> and A/;n C A finite such that for any B € V (A ) the following holds:

//□("*) A/,-n A ->p0 is consistent in C(B) then

any finite subset of □ (“') A ; ->p0 is consistent in C(B).

(3) {B € V(A)|>t £ HSPuC(B)} is a variety.

(4) {B 6 V(A)|^4 £ HSPuC(B)} is closed with respect to ultraproducts.

(5) {B € V(A)|^4 £ HSPuC(B)} is elementary.

If (2) is satisfied by m and A /,», then A/c0 = A((0^m)A/j„ —*• p0)c) and V(A/c0 ) = {B e V(A)|>t t HSPuCA}.Spc (A) := (□{"*) A/jn —*• p0)c is the C-splitting form ula of A above A.

Proo f. Define M := {# € V\A £ HSPfjCB}. We first prove (1) => (3). Suppose that (0 , 0 O) is a splitting-pair in CA. Then the following equivalences hold for ail B € V (A ) : Th(B) D 0o iff Th(C (B )) D 0 O iff Th(C (B )) % Th (C (A )) iff Th(C (B )) % Th (A ) iff

32 2 SUBLATTICES OF AfNA & HSPuCB. W e have shown that V (0 O) = M .The implications (3 ) => (5 ) and (5 ) => (4 ) are trivial. (4 ) => (2 ). Suppose that (2 ) is false.

Then for any m € A 'C A finite there is an algebra # (mtA ') € V (A ) with

( i ) □ ( m) A ,;*-»p0 consistent in C(B(m ) ,

( ii ) C(B(miA')) (= d W A " —► Po for some n and A " C A finite.

First note that, by Theorem 2.4.12, A HSPuCB(m ^ , because □ ( n) A ,,;-«p0 is not

consistent in CB(m t) for some n and A ,; C A finite. Define

K := \J{CB{m%u)\m € w, A ' C A finite } .

A € HSPu(K ) because, by ( i ) , any finite subset o f D ^ A ; - i p 0 is consistent in K and so

we can apply Theorem 2.4.12. W e have shown that K C M but Pu( K ) % M .W e now show (2 ) => (1 ). This is done by proving that (0 , A ((D ^ m^A/in —► p0)c)) is a splitting pair o f CA . Consider a logic 0 i € CA . Then there exists an algebra B €

V (A ) with Th(C(B)) = 0 i . N ow Th(C(B)) 2 A ((D < m) A /in - Po) c) iff d H a jin^Po is

consistent in C(B) iff (by condition (2 ) ) any finite subset o f A ; -*p0 is consistent in

C(B) iff (by Theorem 2.4.12) Th{A) D Th(C(B)) iff Th(C(A)) D Th(C(B)). H

If A splits the lattice CA , then we have the following picture o f CA :

For 4> € C let <f>d denote the dual C -form ula o f 4>. Then (2 ) o f the Splitting-Theorem can

be reform ulated:

(2 ’) There is an m € u) and A / jn C A finite such that for all B € V (A ) the following holds:

I f ( □ ( m)A / tn A -»p 0) rf is consistent in B, then for all n € cj and A ' C A finite ( □ { n) A /A -ip 0) rf is consistent in B.

C o r o l l a r y 2 .4 .15 Suppose that A, 0 , A and A satisfy the conditions of Theorem 2.4 J4 with A finite and A m-transitive for an m £ u. Then

1. 0 splits CA.

2. A / c0 = A ((C3(m)A - pof).

2.4 Splittings in Modal Logic 333. V(A/c0 ) = { ß e V(A)|A i HSCB}.

One might be tempted to conclude that even for not weakly transitive logics A we have V (A /A) = {B\A HSB} if A splits A. For the standard modal logics this is true but itis not generally true. There is a tense logical counterexample in K racht [92]. In Section 2.4.3 we will give a simple monomodal counterexample. In many cases only finite algebras induce a C-splitting:

Proposition 2.4.16 Let C be a good operation and suppose that A has the fmp and 0 splits C A. Then 0 = Th(CA) for a finite and subdirectly irreducible algebra A.

Proof. By Prop. 2.4.6, there is a subdirectly irreducible algebra A with 0 = Th(C (A )). A has the fmp and therefore A = P|{FA(.4i)|i G /} with Ai finite for all i € I. Then A = p){77i(C’(.4,))|i € /}, because A is a C-logic. By definition of a splitting-pair there is an Ai with A/c0 g Th{C(Ai)). But then Th(C (A i)) C Th(C {A )) C Th{A). Now A is s.i. and therefore A G H S Pu{C(Ai)). It follows that A is finite because C{A%) is a finite set of finite algebras.H

For a finite frame g with root 0 reserve a propositional variable py for y G g. The following formula Ag is a well-known diagram of g+.

= /\(Pv -*■ <>iPz\y < « z)

A /\(Pv -*■ ~'Oipz\y4iz)

A A (Pv - 'P z lv t z )

a \J(py\y € g)

It follows that any finite subdirectly irreducible n-modal algebra is finitely presentable with respect to Kn. o := g — {0 } is an opremum of g+ and therefore pa corresponds to ->po in the notation of the definition above. If g C-splits A then the C-splitting formula is SpcA(g) = —*• ->po)c for an m G w.

2.4.3 A Counterexam ple

Define A := K.Altz{Op —► p;p —► DOp). It is easy to check that g € F r (A ) if and only if g is symmetrical, reflexive and no point in g has more than 3 successors.

Theorem 2.4.17 h := | 1 splits A but V (A /h) ^ {B G V(A)|A+ ft HSB}.

Proo f. We first show that h splits A. Denote the two points of h by x and y. Then h splits A if the following holds for all (7 G R fr(A ):

34 2 SUBLATTICES OF AfNA /, A px is consistent in Q iff ü M A/, A px is consistent in Q for all » € w.

By Theorem 1.5.7 any extension of A is Ä-persistent and therefore this follows if we have for all rooted frames g G F r (A ) : A/, A px is consistent in g iff A^; px is consistent in g. The only rooted A-frame in which A^ A px is not consistent is [T ] . The other rooted A-frames devide into four types:

1. Bn > 1 : g = (n, { ( i , j)||t - j\ < 1}).

2. Bn > 1 : g = (n, {(* , j)||t - i| < 1} U { ( n - 1,0), (0 ,n - 1 )}).

3. g = (w ,{(i,j)| | t- j| < 1}).

4. g = {Z , {(i,;)||i — j\ < 1 }), where Z denotes the set of integers.

If g is of type 1,3 or 4 or of type 2 with n = 2m define a valuation on g by ß(px) := {2t|i € Z} n g and ß{py) := g - ß(px).

It is easy to check that g,ß \= A/, and g,ß, 0 f= px. For the case that g is of type 2 with n = 2m + 1 define ß(px) := ( {0 } U {2t + l|t € « } ) n g and ß(py) := g - ß(px). Again g,ß \= A/, and g,ß,0\= px. Notice that it follows that for g 6 F r(A ) : g+ 6 V (A/h) iff h+ ^ HSg+. We now show that there exists B € V (A ) with h+ £ HSB but B £ V (A/h). Define Q := {u, {(* , j')||t — j\ < 1}, {a C u\a finite or cofinite } ) .It is clear that Q+ V (A /h). Suppose that h+ 6 HS&+. Then there is a valuation ß of Q such that /?(□(“ ) A/ijp*) has the finite intersection property. For such a valuation ß either ß(px) is finite and ß(py) is cofinite or ß(px) is cofinite and ß{py) is finite. Hence there exists an m e u> with /?(□"*(pz —* Opy A py —* Op*)) = 0. A contradiction. H

2.4.4 The Use of Splittings

Most of the theorems of this section are not new. They are scattered in the literature of modal logic (consult Kracht [90a],[93], Zakharyaschev [87], Rautenberg [77],[80], Fine [74a],[74b],[85] or Blok [78]). What is new is that we have found a way by means of describable operations to formulate and prove them in a uniform way. We illustrate the theorems by simple examples; more elaborated applications will follow in the next chapters.

Definition 2.4.18 A C-logic 0 € FA is strictly C-one-axiomatizable above A if for any set T of formulas with A (T ) = 0 there exists <f> € T with A (<f>c) = 0 .

The following proposition follows immediately from the results of M cK enzie [72].

Proposition 2.4.19 A C-logic © is strictly C-one-axiomatizable above A if and only if & is a C-splitting of A.

2.4 Splittings in Modal Logic 35Example. It is shown in B lok [78], that T is not a splitting of K. Therefore there are two formulas <f>i,<l>2 with T = K (4>i,<f>2) and T ^ T ^ K {4>2). But we will showthat T = K/s* [x] and therefore it follows that T = or T = K (4>2 ). It followsthat if T is a union of two logics not equal to T, then one of these logics is not a subframe logic.

Proposition 2.4.20 Suppose that 0 is a finite iterated C-splitting of A by (F^,...Fn) and that C is a good operation. Then A (<f>c) = 0 is decidable

if 0 as well as all 0 i € F := Fi U ... U F2 are decidable.

Proof. Consider a formula <f>. <t>c is computable because C is good. Now A(4>c) = 0 iff A (<t>c) D 0 and <f>c € 0 iff (by Prop. 2.4.4) A (<t>c) % 0 i for all 0 i € F and <j>c € 0 iff <f>c & 0 i for all 0 i € F and <f>c € 0 , which is decidable.H

Example. It is decidable whether T = K(<j>s*) because T = K/Si [xjand T is decidable. We remark that it does not follow that it is decidable whether T = because it maynot be decidable whether (j> and <f>s* are deductively equivalent.

Following Zakharyaschev [87] we call a set of formulas V complete for CA if

1. For any logic 0 € CA there is a subset T' of T with 0 = A(r').

2. For all r C T : A(r') € CA.

A set of formulas T is a basis for CA if it is complete and no proper subset is complete.

Proposition 2.4.21 CA has a basis if and only if any logic in CA is a union-C-splitting of A.

Proof. Without restriction we may assume that any set of formulas complete for CA is of the form Tc. Call a formula <f>c indecomposable if there exist no two formulas <j>\ and <j>2 with A (<f>c) = A(<f>\) V A{4>2). By Prop. 2.4.19, a formula 4>c is indecomposable if and only if A (</>c) is a C-splitting of A. Now if all logics in CA are union C-splittings of A then the set of C-splitting-formulas is a basis of CA. But if there is a logic which is not a union-C-splitting then any complete set Tc for CA contains a decomposable formula <f>c and therefore r c \ {<f>c} remains complete.H

Exam ple. £K.Alt\ has no basis: We show that Th( [ T ] ) is not a union-splitting of K.Alt\. In Bellissima [88] it is shown that K.Alt\ is complete with respect to finite and cycle free frames. It follows that only Unite, cycle free rooted frames induce a splitting of K.Alt\. Now consider 0 := K.Alt\/F with F a set of finite, cycle free rooted frames. Then | x—>-*| € Fr(Q ) and therefore 0 ^ Th( [T ] ).

36 2 SUBLATTICES OF AfNDefinition 2.4.22 A partial ordered set (D, < ) without infinite strictly descending chains is a well partial ordered set, a w.p.o. for short, if any set N C D of mutually incomparable elements is finite.

Now suppose that C is a good operation on V (A ). Define a relation <c on V/.S.,(A) by A <c B iff -A € HSCB. It is easy to check that <c is a partial order without infinite descending chains. For a set K C V/.S.,(A) let mc{K ) denote the set of -^c minimal algebras in K .

Definition 2.4.23 Let C be a describable operation on V(A) and 0 € CA. Then 0i is a lower cover of 0 in CA if there is no logic 02 € CA with 0i C 02 C 0-

Proposition 2.4.24 Let C be a good operation on V(A) and let 0 = A/CF be an iterated C-splitting of A by a set F of finite algebras. Then a logic 0i is a lower cover of 0 in CA if and only if there is an A £ mc(F) with 0 i = 0 D Th(C (A )).

It follows that an iterated splitting of CA by finite algebras is finitely axiomatizable above A if and only if it has only finitely many lower covers in CA. Another consequence is that iterated splittings of CA by finite algebras have not more than Ho lower covers in CA.

Exam ple. We have T = K/sf [ x ] . Therefore T n Th{ [~x~|) is the only lower cover of T in the lattice of subframe logics.

We now define one of the main concepts of this essay. The Fine-spectrum of a logic was introduced in F ine [74b]. Here we relativize this concept to compactness preserving sublattices of Afn.

Definition 2.4.25 Let C be a describable operation on V(A) and 0 6 CA. Then the C-Fine-spectrum of 0 above A is defined by

FnCA(Q) := {© i € CA|Fr(0) = F r (0 ,)}.

The fmp-C-Fine-spectrum of 0 above A is

fF n ch iß ) := {0 i € CA|Fr,(0) = Frj{Q x)}.

The C-degree of incompleteness o/0 above A is defined by Cdg^Q) := |F»ca(0)|- 0 is called strictly C-complete above A if Cdg^Q') = 1. The C-degree of fmp of 0 above A is defined by Cdp/A(0 ) := |/FnCA(0)|- 0 has strict C-fmp above A if Cdgf\(Q) = 1.


P rop os it ion 2.4.26 Suppose that C is a good operation on V (A ) and that 0 an iterated C-splitting of A by finite algebras. Then 0 is strictly C-complete above A if and only if 0

is complete, and 0 has strict C-fmp above A if and only if 0 has the fmp.

P ro o f. The implications from right to left are trivial. Suppose that 0 = A/CF is an iterated C-splitting by a set F of finite algebras. Consider 0 j £ C A with F r (0 ) = F r (0 i ) .

Then © i C 0 and V / (0 ) = V / (0 !). Take B £ V (0 ! ) . W e have A 0 H S P u C B for A £ F

because these A are not in V / (0 ). But then B £ V (0 ) and therefore 0 i D 0 . That from

fmp follows strict C-fmp can be proved analogously. H

E xam p le . T has strict 5/-fmp. That means that any subframe logic 0 whose finite

frames are exactly the reflexive frames is T. By a result of B lo k [78] there are 2K° logics whose finite frames are exactly the reflexive frames. It follows that none of these logics is a subframe logic.

P roposition 2.4.27 Suppose that A has the fmp and is m-transitive and suppose that C is a good operation on V (A ). Then the following conditions are equivalent:

(1 ) Any union-C-splitting of C A has fmp.

(2 ) Any 0 £ C A has fmp.

(S ) Any 0 £ C A is a union-C-splitting of CA .

(4 ) CA has a basis.

P ro o f. The equivalence of (3 ) and (4 ) was shown in Prop. 2.4.21. For 0 £ C A define

K © := (V (A ) - V ( 0 ) ) / . 5.t. By m-transitivity of A all algebras in K © C-split A. Therefore A/CK © is well-defined. It is straightforward to check that 0 is a union C-splitting if and

only if 0 = A /CK © .

For the implication from (1 ) to (3 ) suppose that there is a logic Q £ C A with 0 ^

A/CK © . Then 0 D A/CK © and V/.5.,-.(0) = A /CK © ). Hence A /CK © cannot havethe fmp. For (3 ) =» (2 ) suppose that 0 = A/CK © for all 0 £ CA . Then, by Prop. 2.4.26 , a C-logic has fmp iff it has strict C-fmp above A. Suppose there is a C-logic. 0 without fmp. Then T A (V / (0 )) is a C-logic with fmp and the same finite algebras as 0 . A contradiction. The direction from (2 ) to (1 ) is trivial.H

E xam p les . EG, EGrz and ZK4.Z do not have a basis because G , Grz and K4.Z have fmp

but have an extension which does not have fmp. SS4.Z and £K b have a basis because

any extension of these logics has fmp ( B u ll [66] and Nagle & Thomason [85].) The

condition that A is m-transitive cannot be omitted: By a result of B l o k [78] each unionsplitting of K has fmp but clearly not every monomodal logic has fmp.

P ro p o s it io n 2.4.28 Suppose that one of the conditions of Prop. 2.4-27 is satisfied by A. Then the following conditions are equivalent:

38 2 SUBLATTICES OF AfN(1) d c ** o W.p.O..

(2) Any © € CA is finitely axiomatizable above A.

(3) Any 0 € CA is decidable.

(4) CA is denumerable.

Proo f. Let K C V/.S.,(A). It is easy to check that A/CK = A/cm c(K ). Therefore we have for K i ,K 2 C V/.,.t.(A ) that A/cK i = A/CK 2 if and only if m c (K i) = m p (K 2). Suppose that <c is a w.p.o.. Any 0 € CA is representable as 0 = A/CK© = A/cm c(K © ) (K© defined as in the proof of Prop. 2.4.27). But then 0 = A({5p^(.4)|.4 € m c'(K© )}). Now 77ic(K©) is finite and therefore 0 is finitely axiomatizable above A. (2) =» (3) follows from fmp of all extensions of A and (3) =£• (4) is trivial. For (4) =£• (1) suppose that < c is not a w.p.o.. Take an infinite set K of mutually incomparable elements. Then A/cK j ^ A/CK 2 for K i ,K 2 C K with K j ^ K 2. It follows that CA has cardinality 2N°. H

Defin ition 2.4.29 Let C be a describable operation on V (A ) and 0 € CA. The C- spectrum of 0 above A is defined by

•S’pca(Ö) := {©1 € £A|0iTc= © tc}-

Proposition 2.4.30 Let C be a good operation on V (A ) and let 0 = A/CK be an iterated C-splitting of A by a set of finite algebras K . Then

( 1) Spca(G) = {© i € £A|0 D 0 , and A $ V (© ,) for A € mc (K ) } .

If m c (K ) defines an iterated splitting of € A, then

(2) 5Pca(0 ) = [A/mc (K ) ,0 ] .

P roo f. (1) Let © j € Spc a (© ) aQd suppose that A € V (0 i ) for an A € m c(F ). We have C{A) — {.4 } C V (0 ) and therefore C{A) C V (0 i) . But then C (A ) C V (0 j f c) and 0 i t c^ ©• We have a contradiction. Suppose that 0 i C 0 and A £ V (0 i ) for A € mc (K ). Let B € V (0 , Tc)- Then C(B) C V (0 i ) and therefore A t V (C (H )) for A € t » c (K ) . Hence B € V (0 ). (2 ) follows from (1) and the Splitting-Theorem.H

Exam ple. It is proved in Rauten berg [79] that K (O T ) — K/{x |.Hence from T = K/SJ [x] follows SpspdT) = [ ^ (O T ) , ^ .

2.4 Splittings in Modal Logic

2.4.5 Some Results o f W .J .B lok

39

The deepest results concerning the structure of the lattice Af of monomodal logics were attained by W . J. B lok in his articles Blok [78],[80a] . We state some of these results here in order to clarify (1) the complexity of the lattice Af, (2) the need for investigations of proper sublattices of Af and (3) to contrast these negative results with the results concerning the lattice of monomodal subframe logics SAf of the next chapters.

Theorem 2.4.31 A finite rooted frame splits Af if and only if it is cycle free. A logic A is strictly complete if and only if it is a union-splitting of Af. Otherwise A has degree of incompleteness 2**° and has 21*0 lower covers in Af.

Theorem 2.4.32 Any logic A D T, A ^ C, has degree of incompleteness 2H° above T. £T contains only one splitting logic, namely C.

A logic A is tabular if A = Th(F ) for a finite set F of finite frames.

Corollary 2.4.33 Any tabular logic not equal to C has degree of incompleteness 2K°.

Corollary 2.4.34 Any subframe logic not equal to K ,C has degree of incompleteness 2«° and has 2N° lower covers in Af.

Proof. Suppose that A = K/F with F ^ 0 a set of rooted, cycle free and finite frames. Take Q = (g, < ) G F. Then Q £ F r(A ). Now take a point x £ g and define

H := (p U {x } , .Clearly H G F r (A ) and Q G Sf(fH). Hence A is not a subframe logicH

40 3 SUBFRAME LOGICS3 Subframe Logics

3.1 G enera l P ro p ertie s

Lemma 3.1.1 Suppose that g,ß,x |= T. Then there is a countable elementary substructure h of g with x £ h and a valuation a with h, a, x (= T.

Proof. The technique of the proof we give is well-known from F ine [75b]. Suppose that {g, (<«'b < n)),/3,x [= T. Take the first order language with =, and symbols <J, for i < n and D{ for i £ u. Define a structure V = (g, < n), (Di\i £ u )) by £>,- := ß(pi). Nowthere exists a countable elementary substructure (h, (<J, Cl h2\i < n),{hf\ D{\i £ u>)) of V with x £ h. Define a valuation a on h by a(pt) := h n D,. It is easy to show that h, a, x ^ r. H

Proposition 3.1.2 If A is complete with respect to a class of frames closed under elementary substructures, then A is complete with respect to countable frames. A complete subframe logic is complete with respect to countable frames.

Consider a countable frame (g, ( < tj* < n)). Reserve a variable py for any y £ g. We define a set of formulas Xg by

Xg — {pv —* Ofp2|y <ii z}

U{p„ -*■ di-,pz\yftiz}U{pv — ->Pz\y # z}.

Define V s := f\Xg if g is finite. A frame H is subreducible onto a frame g if g is a p-morphic image of a subframe of H.

Lemma 3.1.3 ( 1) A Kripke-frame h is subreducible onto a frame g with root 0 if and only if there exists y £ h and a valuation ß with h,/3, y (= □ (u,).XJ;po.(2) Let g be finite and Ti € Rfr. Then TL is subreducible onto g if and only if there is a valuation ß and y £ h with Tt,ß ^ Vg and H,ß,y po.

Proof. (1) Suppose that / C h and p : / — ► g is a surjective p-morphism. Define a valuation on h by ß(px) := p“ 1 ) for x £ g and take a y £ p-1(0)* I* is easy to stow thatA,/?,y|=ClMX9;po.Now suppose that h,ß,y (= a H ^ jp o - Define b := T rh(y) n U{/3(P*)I* 6 p } and p : hb — ► g by p(z) = x if z £ ß{px). It follows by the definition of Xg that p is well- defined and a surjective p-morphism. (2) follows immediately from the proof of (1) and the fact that Xg is finite if g is finite. H

3.1 General Properties 41

For the proof of the following theorem, which is an extension of a theorem for logics above

K4 in Fine [85], we need a weak version of a theorem proved in Sambin & Vaccaro [88].

For a frame Q = (g , A ) let denote g.

L e m m a 3.1.4 For a modal logic A and Q = (g , A ) £ Rfr{A ) : g C ( ^ a ( ^ ) ) ö for K := 1^1-

P ro o f . There is a surjective homomorphism ß : F r A( * ) — ► Define s : g — ► C^a (*0 )h

by s (x ) := {</> £ C K\x £ ß(<t>) } . 5 is an embedding:

x — y O Va 6 4 : x £ a O y £ a

v<t>e£K: xe ß(<t>) y e ß(<t>)s (x ) = s (y )

x < t y iff s (x ) < t is considered analogously.H

A class of Kripke-frames F has the fin ite e m b e d d in g p ro p e r ty if the following holds

for all g G F r : g 6 F if and only if any finite subframe / of g is in F .

T h e o re m 3.1.5 For a subframe logic A the following conditions are equivalent:(1 ) F r (A ) is universal and A is complete.(2 ) F r (A ) is elementary and A is complete.(3 ) A is D -persistent.(4 ) A is R-persistent.(5 ) A is complex.(6 ) A is u-complex.(7 ) A is compact.(8 ) F r {A ) has the finite embedding property and A is complete.

P ro o f . (1 ) => (2) is clear and (2) => (3 ) is shown in Fine [75b]. For the implication from

(3 ) to (4 ) suppose that (y, A) £ Rfr(A ). Then, by Lem m a 3.1.4 , we have g □ (^*a ( « ) ) jj

for k := |j4|. But ( ^ a ( ^ ) ) ji is in F r (A ) because A is £>-persistent. (4 ) => (5 ) and (5 ) ^(6 ) are trivial. (6 ) => (7 ) follows from Theorem 1.4.1. For (7 ) implies (8 ) suppose that

there exists a frame g £ F r (A ) such that all finite subframes are in F r (A ) . By Prop.3.1.2

we may assume that g is countable and has root 0. Now any finite subset of O ^ X g;po is

A-consistent because these sets are consistent in finite subframes of g. Hence D ^ X g\po is A-consistent. It follows that A is not compact: For otherwise there exist a A-fram e

/i,y £ h and a valuation ß with hyß yy f= O ^ X g;po and g would be a A-fram e, by Lemma

3.1.3. The implication from (8) to (1) is proved in Tarski [54]. H

Corollary 3.1.6 (Goldblatt) G is not complex.

Proof. G is a subframe logic and not elementary. Therefore G is not complex.H

One should compare this proof with the proof in G O LD B LA T T [89] to recognize the

power of the theory of subframe-logics. A logic A is basic elementary if there is a finite

42 3 SUBFRAME LOGICSset of first order formulas T such that for h € Fr : h € F r(A ) if and only if h f= T. For a finite rooted frame g let Pg denote the universal formula in first order logic with h f= Pg iff g % h for all h € Fr.

Corollary 3.1.7 Suppose that a subframe logic A is finitely axiomatizable, complete and elementary. Then there is a finite set of finite rooted frames F such that for all h € Fr : h € F r (A ) iff h ^ Pg for g £ F.

P roo f. In Van B enthem [83] it is proved that any finitely axiomatizable, elementary and complete logic is basic elementary. Now Fr{A) is universal and therefore there exists a finite set of universal formulas T such that h € F r (A) iff h )= T. H

The modal axiomatic universal classes of Kripke frames have a simple characterization:

Theorem 3.1.8 A universal class of Kripke frames F is modal axiomatic, i.c. there exist a modal logic A with F r (A) = F, if and only if F is closed with respect to p-morphic images and disjoint unions.

Proo f. The direction from left to right is trivial. For the other direction suppose that g € F r(Th (F )). Then, by Lemma 3.1.4, we have g C (^r/»(F)(K))ll f ° r K := 2^1. Therefore g € F follows if (^r/i(F)(K))ll € F. But, by a result of FINE [75b], (^rfc(F)(K))lt is a p-morphic image of an ultraproduct of a disjoint union of frames in F. -I

Corollary 3.1.9 For a set of finite rooted frames F the class K := {h € Fr\f % h for f € F }

is modal axiomatic iff K is closed under p-morphic images.

Proo f. K is a universal class closed under disjoint unions.-i

3.1.1 5/-splitting-formulas

The lattice of n-modal subframe logics is denoted by SAfn and for an n-modal subframe logic A define SA := £A n SAfn.For n € w we know that Sf is a describable operation on M A n. Hence the splitting theorem tells us how to determine the splittings of a lattice SA. Nevertheless it will pay off in the next chapters to have a closer look at the special situation of splittings of lattices of subframe logics. For a finite frame Q with root x let S fm(^ ) := (□ (”*)Ag —► ~'Px)Si ■ It follows from the splitting theorem that Q splits SA if and only if there is an m € w such that A (S fm((?)) = A (S f“ ({7)) for all n > m. In this case we have A/S*G = A (S fm(^ )). F ine [85] defines subframe logics above KA as follows. A logic A above KA is a subframe logic if and only if there is a set F of finite and rooted A 4-frames such that

A = Ä ’4({II](1)V0 —*■ -*Px\G € F ,x a root o f G}).

3.1 General Properties 43So far we do not know whether —* ->px is deductively equivalent to S f1(^ ) aboveK4. First of all it is not the case that for all finite Q with root x and m € u> the formulas S fm(£ ) and - -ipx are deductively equivalent: Consider the frame

Let < ? := [• ]. It is readily checked that h ^ S f2(£ ) but that h (= —► ->px. Thefollowing lemma clarifies the situation.

Lemma 3.1.10 Let H £ Gfr,m £ u> and Q £ F rj with root x. Then

( 1) D ^ A s A px is consistent in H=> D(m)v^ A px is consistent in H => A px) j q var(V^), is consistent in H

(a (m)A£ A Px) 1 q, q & var(Ag), is consistent in H.

(2) *„(□("*>Ag - ^px) C ffn(D<m>V0 - - P l ) C Kn(a ^ )V g - ^px)sS) = ff« (S f“ (G)).

Proo f. (1) We show that (□ (”*) V$ A px) j q consistent in TL implies (□ (”*)A g A px) | q consistent in H. The other implications follow from A g = V g A \/{Py\y € g). Consider a valuation ß and z £ h with H,ß, z (□ ("*)V$ A px) j q. Define

7(? ) := ß(q) n ß(\f{py\y e 5) ) and 7(pv) := ß(pv) for y € g.It is easy to check that W ,7 ,z (= (□ (TO)A{; A px) f q. (2) follows immediately from (1 )H

Proposition 3.1.11 Let H be refined and let all frames in Sf(H ) be n-transitive. Further suppose that Q is a finite frame with root x. Then the following conditions are equivalent:( 1) □ (’0 v p A px is consistent in 7i.(2) (o (n)A(? a px) 1 q, q £ var(Ag), is consistent in H.(3) There exist sets by € A,y £ g, with V^[6v|y £ g] = h and bx ^ 0.(4) Q € Fr(S f(H )).(5) H is subreducible onto Q.

Proo f. (2) (4) follows from Corollary 2.4.13. (3) => (1) => (2) are clear. (3) O (5)follows from Lemma 3.1.3. We show that (2) implies (3).Suppose that H,ß,z (= ( o M A j? A px) f q. Define by := ß(p (n)Ag J. q) D ß(py) for y £ g. It is readily checked that {by\y £ g } satisfies (3).H

By duality, we get from the equivalence of (4) and (5) that for all finite subdirectly irreducible algebras A and algebras B £ V{K.Trn) the following equivalences hold:

A 6 V(Sf(B )) iff A € H S(Sf(B )) iff A € S(Sf(B )).In other words, we do not need homomorphisms to generate the finite and subdirectly irreducible algebras in n-transitive varieties. If follows that

V{K.Trn/sU ) = {B £ V (K .Trn)\A i S (S f(B ))}.

44 3 SUBFRAME LOGICSCorollary 3.1.12 For all finite monomodal frames Q with root x and n > 0 we have

Ä\Trn(S f“ (a ) ) = K.Trn(a l* )V c ^px).

Hence above K4 F ine ’s subframe formulas O^'Vg —*• ->px are deductively equivalent to the S/-splitting formulas S f1(^ ). We will use the results of this section without referring to the propositions.

3.1.2 Subframe Logics and Confinal Subframe Logics above K 4

Theorem 3.1.13 The subframe logics above K4 are precisely the subframe logics defined in F ine [85]. Therefore any subframe logic above K4 has fmp and is a union-Sf-splitting of K4 and SK4 has a basis.

Proo f. We denote the set of subframe logics defined by K. F ine by V. We know from Corollary 3.1.12 that a logic 0 is in V if and only if there is a set of finite rooted frames F with 0 = A '4 ({S f1(^)|^ G F }) . It follows that SK4 D V and that V is the set of union-S/-splittings of K4. By F ine [85] all logics in V have fmp and therefore all union- S/-splittings of K4 have fmp. By Prop. 2.4.27, all logics in SK4 are union-5/-splittings of K4. Hence SK4 = V. By Prop. 2.4.21, SK4 has a basis.-!

The similarity between the subframe-formulas introduced in Fine [85] and frame- formulas introduced in F ine [74a] is completely explained: Both of them are splitting- formulas; the frame-formulas are splitting-formulas for ZK 4 and the subframe-formulas are splitting-formulas for S K 4.

We denote the operation Sfc<m by Cf. C f is good a operation on Gfr(K4) and the C/-logics are exactly the confinal subframe logics defined by M. Zakharyaschev. Za- kharyaschev associates with a finite transitive and rooted frame Q a formula a((/,0, _L) such that for all H e Gfr(K4): 6 € Fr{C f(H )) iff H £ o(0 ,0 ,-L ). It follows that K4/clG = K4(-ya(G,9, L ) ) and that ->a(G,9, J.) and (D ^ A g —*• ->po)C* are deductively equivalent above K4. Hence the canonical formulas ->a(£,0, ± ) are, up to deductive equivalence, the C'/-splitting-formulas. For a finite transitive frame G with root x we have the following situation:

(:Th(G), K4(aU) a c —► ->px) ) is a splitting-pair in ZK4.

(Th {C f(G )),K 4{{pU)&g -,pr )<?/)) }s a splitting pair in CfK4.

(T h (S f(G )),K 4((dU )^ß — ->px)sf ) ) is a splitting-pair in SK4.

It is shown in Zakharyaschev [92] that all confinal subframe logics are union-C/- splittings oi K 4 and therefore have the finite model property. A lot of consequences follow immediately from the propositions we proved in The Use of Splittings. For instance it follows that S p c j k a { A ) is always an interval. For let A = K4/clF . Then it follows from Proposition 2.4.30 that Spcjka{A) = [Ar4/mc/(F), A].

453.2 Basic Splittings of SAfn

3.2 Basic S p littin gs o f SAfn

Given two subframe logics 0 ,A with 0 € SA there are mainly two questions we try to answer in the following chapters:

(1) Is 0 an iterated 5/-splitting of A by finite frames?

(2) Is 0 strictly Sf-complete above A?

By Proposition 2.4.26, we know that a positive answer for (1) implies a positive answer for (2) if 0 is complete. If 0 is Ä-persistent and A is m-transitive even the converse direction holds: For suppose that 0 is Ä-persistent. Then, by the finite embedding property of Fr(Q), there exists a set of finite rooted 0-frames F such that

Fr(Q ) = {h e Fr(A)\G % h for Q e F }.But then 0 V S fm(£ ) for G € F. Define 0 i := A ( {S fm(£)|£ € F }) . Now it is easy to check that F r (6 ) = F r(Q i) and it follows that 0 is strictly ^/-complete above A if and only if 0 = ©i if and only if 0 = A/^F. We conjecture that this is also true for not weakly transitive logics but have found no way to prove this.

Definition 3.2.1 Suppose that F = (/, (rj|t < « ) ) and Q — (g, (s.jt < » ) ) are finite n- frames and x £ g. Then F is an x-arrow subframe of Q, F <x G, if f = g, r, C s, for i < n, and x is a root of T . T is a strict x-arrow subframe of Q, in symbols T <x Q, if F <xG and F ± Q .

We explain this definition by an example. Consider the frame G = | x ^ Let x be thereflexive point and y be the irreflexive point in G- Then the frame F = •-—x| is not an x-arrow subframe of Q but is a y-arrow subframe of G- The reason is that x is not a root of F.

Troughout the following three chapters F (G) denotes a finite frame (/, (r,|i < « »( (9,(s ,jt < n ))) with root 0. In the monomodal case we write and r (s) instead of rj (s j). We write ( 1C —* G) if G is a p-morphic image of K and (F —► G) if F is a set of frames such that for any IC € F the frame G is a p-morphic image of 1C.

Definition 3.2.2 Suppose that 7f = (h, (<li|t < n), A ) is a refined n-frame and suppose that b C h is finite such that hb is rooted. Further suppose that (Aj —*• G) via a p-morphism p and F < 0 G- In this case we write (F < 0 G « - hb). A set {a v|y € g} C A recognizes (F <o G <— hb) in H if

(R £) Fory eg : p-1(y ) C ay.

(RO) For y,z e g : ay Oiaz = h if y ri z.

(ÄO j For y,z e g . ay —> az h if y z.

46 3 SUBFRAME LOGICS(Rn) For y,z e g : ay C\az = 9 i fy ^ z .

(F <o Q *— hb) is recognizable in 'H if there exists a set which recognizes (F <o Q *— hb) in H. We write

(G - hb) if F = G ,(F <o G — hb) if G - hb,

(G * hb) if G — F and Q a hb,(F <o h ) if G — hb.

Note that we allow b to be not internal bnt that we do not allow b to be infinite. Let H = (h, A) G Rfr and let Q be finite with root 0. We have (1) =» (2), where

(1) There is a finite set 6 C h such that (Q <— hb) is recognizable in H.

(2) There is a set {ax\x G </} C A with Vg[ax\x G g) = h and ao ^ 0.

Let Q = (g, (s,|i < n)). Suppose that {ax\x € fir} recognizes (Q <— hb) = (6 <o 6 <- hb) in H. Then r* = s, in (RO ) and Vg[ax\x G g]=- h follows from the definition of V^. Further a o 2 p " 1(O )# 0 .

The main tool for proving that a frame Q 5/-splits a logic A will be the following

Observation 3.2.3 Suppose that A is an R-persistent subframe logic and Q G Fr/(A) is rooted. Then Q Sf-splits A and A/S*Q is R-persistent if there is a finite set of finite rooted frames Fg C Fr(A ) with {Fg —*■ Q) such that

R I There exists anm £w such that for all h G F r( A ) : If d (to)V<; A po is consistent in h, then there exists AC G Fg with IC C h.

R II YH € Rfr{A ) and AC € Fg : If AC a hb for a set b C h then (G ♦— hb) is recognizable in H.

In this case Fr{A/s-fG) = {h G Fr(A)|AC % h for K G Fg) andA/s'g = ACSf“ ^ ) ) = A(D("*>VC - -po).

P roo f. Take an m G u> which satisfies (R I) and suppose that (□("*) Ap A po) } q is consistent in H G Rfr (A). Then there is a subframe H \Q H with Apo consistentin Ji\. By (R I) there is a AC G Fg with AC C h. Hence, by (RH ) and the implication (1) => (2) above there is a set {a x|z G g ) C A with Vg[ax\x £ g] = h and ao # 0. But then (□ (n)Ap A po) i q is consistent in H for all n £ u>. It follows from the splitting theorem that Q 5/-splits A. The other claims follow immediately with Lemma 3.1.10.H

We need some more technical notations:

47

Definition 3.2.4 For a finite T-cycle free frame Q = (g, (stji < n)) define the depth of x e g in Q by

dpg(x) = 0 Vj/ £ Trg(x) : x = y.dpg(x) = A: + 1 3y £ Trg(x) : dpg(y) = fc andVy £ Trg(x) : dpg(y) < k o rx = y.

The depth of Q is defined by dp(G) := max{dpg(x)\x £ p }. The inverse depth of x £ g is dp-(x ) := dp(G) - dpg(x).

3.2 Basic Splittings o f SMn

Lemma 3.2.5 (1) Suppose that TL is refined, that (F < 0 G <— hb) and that T is cycle free. Then (F <0 G hb) is recognizable in TL.(2) Suppose that TL £ Rfr((tiuT), that F £ Fr(oonT ) is T-cycle free and that (T <o G * - hb)- Then (F <o G h ) is recognizable in TL.

Proof. (1) Suppose that TL = (h ,A ) is refined and that p : hb — ► G is a surjective

p -m orph ism for a finite b C h such that hb is roo ted . For y £ g P~l(y) is finite and g is

finite; therefore there exists a set {cy\y £ g} C A such that cy D p~l (y ) fo r y £ g and

cj/i ncV2 = 0 fo r 2/i # V2 - For yl A y2 there exists a set d*(j/liJ/2) € A w ith d }yiflo) 2 P~\y2 )

and p ” 1 ( 2/1 ) C N o w define fo r y £ g:

by := Cj/ n A*>* < n} n A jm < « }•

It is easy to check that {by\y £ g} C A satisfies the conditions ( R € ) , (R □ ) and ( R H ).

W e now define a set {ay\y £ g} by induction on dpp(y). Suppose that ay is defined for

dpr(y) < n • T hen define fo r dpp(y) = n:

ay := by H z).

T h is is well-defined because fo r y, z £ g y rt z im plies dpp(y) < dpr(z) since F is cycle

free. W e show that {ay\y £ g} recognizes (F < 0 G <— hb). ( R H ) is obv ious because

ay C by fo r y £ g. (R □ ) fo llow s from ( R □ ) fo r {by\y £ / } . ( R O ) and (R € ) are proved

by induction on dp^(y): Suppose that ( R O ) and ( R € ) a re show n fo r all z £ g w ith

dpp(z) < n and that dpjr(y) = n. I f z\ £ ay and y rt* z then , by defin ition , z\ £ O ta z .

It fo llow s that ay —► O ta - = h. N o w p “ 1(y ) C ay fo llow s fro m p~l {z ) C a z fo r y r±z, by

induction hypothesis.

(2 ) T h e construction is analogous: D e fine {6 y |y £ g } as above . T h en define by induction

on dp?(y) a set { a y|y £ g ) C A :

ay by Pi P|{Ot'az|y ri y ^ z}.

Again this is well defined and (R fl) and (R □ ) follow immediately. For (R O ) and (R £) notice that from Ti £ Rfr(®nT ) follows ay C Otay for all y £ g and i < n. H

48 3 SUBFRAME LOGICSCorollary 3.2.6 (1) Suppose that H G Rfr, that T is cycle free and that ( T *— hb). Then (T *- hb) is recognizable in H.(2) Suppose that Ti G Rfr((ftnT ), that T € FY(®nT) is T-cycle free and (F <— hb).Then (T <— hb) is recognizable in H.

Theorem 3.2.7 (1) A finite rooted n-frame Q Sf-splits Kn iff Q is cycle free. In this case Kn/S}G = Kn(o (M G ))^ g -,po) = K n(S idp °\G)). Any union-Sf-splitting of Kn is R-persistent.(2) A finite rooted frame G G Fr(tonT ) Sf-splits WnT iff G is T-cycle free. In this case WnT/SJG = (><)nT(0^dp V g —» -'Po) = C?DnT (S f<p((?)(ö)). Any union-Sf-splitting of O0’*T is R-persistent.

Proo f. ( 1) Suppose that G G F rj is cycle free. We define a set Fg and check conditions (R I) and (R II) for (G,Fg). First define m(y) G u for y G g by induction on the inverse depth dp~(y) := dpg(y) : m(y) = 1 iff dp~(y) = 0. m (y) = £(Tn(z)|zs,l/,y # z,i < n) if m(z) is defined for all z G g with dp~(z) < dp~(y). Now define

Fg := {K, G Ftj\K. rooted, 3p : K — ► G surjective with Vy G g : |p-1(y)| < m (y)}.

It follows immediately that Fg is a finite set of finite, cycle free rooted frames. By Corollary 3.2.6, (G, Fg) satisfies condition (R II). Define m := dp(G). For (RJ) suppose that h G Ft and h,ß,x (= A po• By induction on Z € v we define sets 6/ C h : bo := {a:}.Suppose that 6/ is defined. For i < n ,y € h and xu X2 € g define

t*2 :== i z € h\z g ß(px2) , y < i z , y e ß(Pxi) ,y £ o*(bt n ß(pX2))}-

Now let at be a set containing exactly one element from any ,*2 ^ 0 and define bi+\ := b[ U Of. Let b := bm. We prove that hb G Fg. It is straightforward to check that p : hb — ► G defined by p (y) := z if y G ß(Pz) is well defined and a surjective p-morphism. |p- 1(z)| < m(z) is proved by induction on dp~(z). I f dp~(z) = 0 then p- 1(z ) = { x } and therefore |p- 1(z)| = 1. Now suppose that |p- 1(zi)| < m (z j) for dp~{z\) < dp~(z) and dp~(z) > 0. By definition of b for y i,jf2 € p- 1(z ) with yi ^ jft there exist i , j < n and x i ,®2 G b with x\ <], yi and *2 < j V2 and t ^ j or i j * 2- There exist zi,Z2 € g with x\ € P- 1( z i ),*2 € p~l (z2). But then zjS.z and z?sjz and therefore dp~(z^),dp~(z2) < dp~(z). It follows that |p- 1(z)| < ^,{m{z\)\z\SiZ,z # z i,t < « )•

For the converse direction suppose that G is not cycle free. By Theorem 1.5.2 Kn = {\{Th{Sf{F))\F a finite n-tree } C Th(Sf(G )).

But T h (S f(F )) % Th(Sf(G )) for any finite n-tree T . This follows from the fact that any p-morphic image of a finite cycle free frame is cycle free. It follows that Th(Sf(G ) ) ‘ is not prime in SK n.

(2) The proof that a finite rooted T-cycle free frame S/-splits ®nT is word by word the same proof as above. That a frame G which is not T-cycle free does not 5/-split <x>"T

3.2 Basic Splittings of SAfn 49follows with Theorem 1.5.2 from

C%nT = f){T/i(5/(.T))|.T a finite reflexive n-tree } C Th{Sf(G)) and Th (S f(F ) ) % Th(Sf(G )) for all finite reflexive n-trees T . This follows from the fact that any p-morphic image of a finite T-cycle free frame is T-cycle free. H

Examples. T = K/Sf\x\and T.B\ = T/Sf{^=^\ = K/Sf\x\ /SJ | ^ 7 | .

Corollary 3.2.8 A finite rooted frame splits Kn if and only if it Sf-splits Kn.

Corollary 3.2.8 does not hold for T because T has only one splitting: (T fiflT] ),£ ).

Corollary 3.2.9 ( 1) Suppose that G is a finite n-tree. Then F r{K n/s G) = {h\G £ h}. (2) For a finite reflexive n-tree G Fr($)nT/s*G) — {h € Fr(oonT)|(y £ h}

Some remarks concerning this proof: That a finite rooted cycle free frame .S/-splits Kn follows from B l o k s result that these frames split Kn. But let us note once more that we cannot conclude that Kn/S G is ß-persistent even if we use the fact that Kn/G is ß-persistent. Nevertheless there is a more elegant proof for the irreflexive case of both facts via unravelling. In this case the set Fq can be defined as the set of p-morphic images K of the total unravelling of G such that G is a p-morphic image of K\ this set is finite because G is cycle free. Such a simple definition is not possible for the reflexive case, even if we use reflexive unravelling. It is in general not possible to construct a set of T-cycle free frames Fq for a T-cycle free frame G such that

(I) Fr(T/SfG) = {h e Fr(T)\K % h for K € Fq}. As an example define G •— I^E G Sf-splits T because G is finite, rooted and T-cycle

which is notfree. But it is easy to prove that any set Fq with (I ) must contain T-cycle free.

Theorem 3.2.10 ( 1) There are 2N° union-Sf-splittings of K and K.AU2. (2) There are 2N° union-Sf-splittings o fT and T.AU3.

Proo f. ( 1) For n e u define Gn = {<7«, rn) by

9n '•= {(0, m)|m < n} U {(1, m)|m < n} U {0, xn, yn, vn, tun}.r n •— {((0j n } i x n )i ((0» w)> J/n)> ((1» n)j vn)j ( ( 1 » ^n)> (0, (0,0)), (0, (1,0))}u = *+ < « } u {((!,*)> (i> i))li = * + i , i < « } .

All Gn are cycle free and therefore 5/-splits K and KJSiM is a union-5/-splitting of K for

50 3 SUBFRAME LOGICSM C {Qn\n 6 w}. We show that A '/^M ^ K/s*N for M N . This follows if Gm is not subreducible onto Qn for m ^ n. Suppose that there exists b C gm with p : (Gm)b — *■ {/„. Then m > n. We may assume that (Gm)b is rooted. Now it follows from the definition of p-morphism that if p(x) has k successors then x has at least k successors. It follows that 0, (0, m), ( l,m ) € b. Now it is obvious that 6 = gm. Hence p is a p-morphism from Gm onto Gn• It follows by induction on dpgm(x ) that dpgm(x ) = dpgn(p(x)) and from this that m = » .All Gn are K.AU2-frames and 5/-split K.Alt2 because they 5/-split K.Again K.Alt2/SiM # K.Alt2/Sf N for M # N .(2) For define Hn = (hn, s„) by hn := gn and sn := rn U {(x,x)|x € hn). All Hn are T-cycle free and reflexive and therefore 5/-split T. The same arguments as above show that T / ^ M ^ T/Sf N for M ^ N and M ,N C {Wn|rt € w }. The claim for T.Alt3 follows immediately. H

Let T = (t, (< t |i < n ))((t, {<,• |t < « ) ) ) always denote a (reflexive) n-tree with root 0.

Definition 3.2.11 For a (reflexive) n-tree T define (T ) := {T € Fr\T <0 F ). Then ( (T ) , < 0) is a finite partially ordered set with minimum T. For T € (T ) the T-closure of F is F - := {fC e (T)\K <0 F ) and the strict T-closure of F is F < := F - — { T } . A set F C (T ) is T-closed if F e F implies F - C F.

Note that the set F - = {K, € (T)|K. <0 F } depends on T . Nevertheless we will often write F - without referring to T , if this causes no confusion.If we try to show that a specific logic is an iterated 5/-splitting it is useful to have a canonical axiomatization of this logic. Such a canonical axiomatization should show the geometrical meaning of the axioms. One possible approach are the sketch omission logics introduced in Kracht [90b]. Indeed, it can be shown that the logics defined in the following proposition are sketch omission logics.

Definition 3.2.12 For a finite n-tree T = ( g, (<,• |t < n)) and T <0 G = (g, (s«|t < » ) ) define

:= A < p* -* “’Pvl1 # y)

A f\{P v — OiP*\y < i *)

A A<Pv — “"O.p^lx foy)

Proposition 3.2.13 For H € Rfr and T <0 G : A po is consistent in Hif and only if there is an F € G- with F Q h. A := A'n(a (<<p(rI)V c< -*■ -.po) j<? an R-persistent subframe logic with F r (A) = {h € Fr\F % h for F 6 G-}-

Proof. Let G = (g, (s,|t < n)). Suppose that h,ß,x\= □<<<p(r ))V i;< Apo. It is easy to show that there exists a frame F with T <0 F and b C h such that p : F — *• defined

51

by p(y) £ ß(py) 0 b is an isomorphism. Then we have for yuV2 € t that p(y\) < , piui) implies y\S{y2 . Hence T <o 6-Now suppose that H £ Rfr(A) and that there is a frame T £ Q - with T ~ for a b C h. By Lemma 3.2.5, (T < 0 T ~ h ) is recognizable in 7i. But then is consistentin 7i and it follows that g<\po is consistent in 7i\ a contradiction. H

It is important to note that we cannot conclude that the logics defined in Proposition 3.2.13 are iterated 5/-splittings of Kn. To the contrary we will show that many of these logics are not iterated 5/-splittings and even not strictly S/-complete. Note that D n^£?< —'- “*Po is deductively equivalent to

□ («) \Z(V?r|T <o T <o Q) —'- ~but it will follow from the results concerning strict 5/-completeness that this formula is not always deductively equivalent to

\/(D(n)V^r|T <o T <o Q) —'- -*Po,even above ?i-transitive logics. (There is a mistake concerning this point in Kracht [90b].) First, it is readily shown that

O r , ( ( D W ^ - ->p0|T < 0 T <0 </})= A-.Trn(V (a (n)Vjr|T < 0 -Po).

As an immediate consequence we get that the following two conditions are equivalent:(1) A\7Vn(DWVfls - “>po)

= K.T r„( V (□^n) V^-|T < 0 -Po)(2) K.Tr,, (□ (’*) V c< —> -ipo) is strictly 5/-complete above K.Trn.

This equivalence shows the subtle difference between axiomatizations by sketch omission formulas of Kracht [90b] and axiomatizations by 5/-split ting-formulas.

3.2 Basic Splittings o f SAfn

Examples (I) For m £ lj define Tm := (m + 1,5), where t S j iff j = t + 1.

Tm = <r*— •x-«x __m

Now define := (m + 1 ,< ) and N m := (7|‘ ) - . It follows from Theorem 3.2.7 that N m defines a union__5/-splitting of K and that K / ^ N m = A '(D (m)'v'isim —► ->p0) is R- persistent. This union-5/-splitting will play a fundamental role in our investigation of monomodal 5/ splittings.

(II) Define Tn := ( « + 1 ,< ) and B „ := Now let Bn := K {D ^ V B- — -.p0). It is readily shown that, up to deductive equivalence, p —* DOp = B\.

(I ll ) Define Kn := (n + 2, {(t ,i)| j < t + 1}) and Tr~ := Kg.

Then K.Trn = ^ ( ü ^ V ^ . - — -.po).

(IV ) Define I n := < ■ and Qn :=^ x n + l ^ •n + l

Define I * := Then K.In = A (D (1>V1- - -,po).

52 3 SUBFRAME LOGICSLet us note two technical corollaries which will be useful for determining 5/-splitting logics.

Corollary 3.2.14 Suppose that Q G (T ). Define M := {h\T £ h for T G G<}- Then for h G M :

G C h if □ (dP(T ))V ß A po is consistent in h.

Hence, if A is an R-persistent subframe logic with F r (A ) D M , then (R I) is satisfied by m = dp(T) and Fg = {G}.

Corollary 3.2.15 Suppose that Ti G Rfr, that T <o T and that {ay\y € t} recognizes {T < 0 G — hb) in Ti. Further suppose that h G M , M defined as in Corollary 3.2.14- Then for any x € a0 there exist b\ C h such that x is a root of hbl and an isomorphism p : G — ► hbi w*th p(0) = x so that {a y|j/ G t j recognizes (F <o G — hbt ) in Ti.

For a frame (g, (<Jj|t < n)) define the transitive closure of (<J,|t < n ) by:(x,y ) G (<l,|i < n)* iff there is a path of length > 0 from x to y.

Defin ition 3.2.16 For an n-tree T the set (T )r C (T ) is defined by :T G (T )r i f fT < 0F and r* C ( < j \j < n)* U {(x,0)|x G t} for i < n.

It follows immediately that ( T )r is T-closed. For T <o G let (?* denote a fixed maximal cycle free frame with T <o G <o G- Then we extend the definition of depth to not cycle free frames G- For y € g define dpg(y) := dpci(y).

Theorem 3.2.17 Suppose that G € (T )r and that A is a R-persistent subframe logic until Fr{A ) C {A|/C £ h for K. € G<}- Then G Sf-splits A and A/S G is R-persistent.

Proo f. Define Fg := {G}- We check (R I) and (RH). (R I) follows from Corollary 3.2.14. For (R II) suppose that Ti € Rfr(A ) and G — hb for a set 6 C h. By Lemma 3.2.5, there exists a set {&y|y € t } which recognizes (T <o G — hb) in Ti. By induction on dpg(y) we define a set {a v|j/ € t} C A. Suppose that ay is defined for dpg(y) < m and that dpg(z) = m. Then

o.z :— bz 0 Pl'fOjfly |xs,*y, y ^ z, y ^ 0, i < n ) n W|0, i ^ r }.

az is well-defined because G G (T )r . We prove that {a y|j/ G t} recognizes (G — hb). First it is easy to show that

(I ) for any &i C k with G a: hbl : {a y|y G t } recognizes (T < 0 G ^ h ) in Ti if {by\y G <} recognizes ( T <o G — hbt ) in Ti.

3.2 Basic Splittings of SAfn 53

For (G ~ h ) and {a y|y € t} the conditions (R€), (R ü ) and (Rfl) follow from the corresponding properties of {by\y 6 t}. For (RO ) suppose that z € ay and y$iyi. If y\ ^ 0 then z € 0 ,ayi, by definition of ay. Now suppose that y\ = 0. Then, by definition of ay, y € Oi&o. Take x € 60 with y < , x. By Corollary 3.2.15, there exists bi C h such that has root x and (T <0 Q ~ h^) and this is recognized by {6y|y € *}. By (I), {ay\y € *} recognizes (T <o G ~ h&J. It follows that x € a0 and therefore that y € O,a0. H

For F € (T ) the height of T in ( (T ), < 0) is defined in the obvious way so that the height of T is 0. Ft(T ) denotes the set of frames of height i in (T ). For F C (T ) closed define (F )i := (F0(T ) H F, F i(T ) n F , . . . , F„(T ) n F ) where n is maximal with F n FW(T ) ^ 0. Now the following corollary follows by induction.

Corollary 3.2.18 Suppose that F C ( T )r is closed. Then (F ), defines an iterated Sf- splitting of A'n.A"n/5f (F )t is R-persistent and F r(A 'n/ ^ (F ), ) = {h € Fr\G g h for G € F }.

We will often abbreviate A /Sf (F ), by A /S F and say that F defines an iterated ^/-splitting of A.

Example. Define an n-tree T by T := ({O },({0 }| i < n)). Then (T )r defines an iterated 5/-splitting of Kn and <X)nT = Kn/ S* (T )r .

Definition 3.2.19 For a reflexive n-tree T = (*,(<,• |t < n )) define a set ( T ) rcj C (T ) by: T € (T)/*c/ iff T € (T ) and rt* C (< t- |i < n)>* U {(x,0)|x 6 *} for all i < n.

The proof of the following theorem is anologous to the proof of Theorem 3.2.17.

Theorem 3.2.20 Let T be a reflexive n-tree, G € (T )R ef and let A be an R-pcrsistcnt subframe logic above (&nT such that F r (A ) C {h\fC g h for K € GK}> Then G SJ-splits A and A /S*G is R-persistent.

Note that any iterated 5/-splitting of ®nT is an iterated 5/-splitting of K n because C*>nT is an iterated 5/-splitting of Kn.

Example. We need some conventions for pictures of n-frames. A n-frame ($r, (<l,|i < n » is drawn as ((g> < t)|z < n).

Define 7u := ( 1 1, 1 • •! ) and Gu •= ( I — |, [ » —• [ ).

Go ^ (Tu) and (T®T ).U\ = T <#T/sfGg- Hence T <#T.Ui and T ® T./d are iterated 5/-splittings of A'2 by finite frames and strictly 5/-complete.

54 4 THE LATTICE OF MONOMODAL SUBFRAME LOGICS4 The Lattice of Monomodal Subframe Logics

Let us start with a list of the results for the monomodal standard systems. The listed properties will be shown in this chapter with the exception of the result that all monomodal subframe logics above K.Altn for an n > 0 (and therefore all tabular subframe logics) are strictly ^/-complete. This will be shown in Chapter 5, where we prove that all polymodal subframe logic above K.Altn are strictly 5/-complete.

Subframe logic A Strictly 5/-completeV er +Triv +

T +□ nJ_ € A +A tabular +

K4 C A C G.3 ___

A'4 C A % G.Z +K.Trn -

K.Bn +K .In —

A D K.Altn +A ’Bm'In +A D K5 +

Iterated 5/-splitting by finite frames++++?

+

+

1

++

4.1 Basic Monomodal Splittings

D efin ition 4.1.1 Let h € F t and b C h. The n-reflexive iteration bn of b in h isdefined as follows: bo := 6,6n+i := bn 0 06n.

Lem m a 4.1.2 Suppose that h € F t and that n £ u with T % h for all T £ N n- Let b* denote the n-reflexive iteration o fbC h . Then b* —*• Ob* = h.

P roo f. Let x € b*. Then there exists a path {x „ <J xn_ j < ]...< ] * 0} Q h with x = xn and x, 6 6, for t < » . Suppose that x ft Ob*. It follows by induction that s, £ 6,+i . Hence x, ^ Xj for * ^ j and x,^J xj for j > t. But then ({x,|t < n } ,< n {x,|t < n }2) € N n, which is a contradiction.-!

CoroUary 4.1.3 Suppose that A is a R-persistent subframe logic. Then |T] Sf -splits A and A/Sf [T] is R-persistent if and only if there is an n £ u with A D K/s*N n.

P roo f. Suppose that [T| 5/-splits A and Ai := A/S} [T )is ^-persistent. Let T\ denote the first order theory of Fr(A\) and suppose that for n € w there exist gn € N n with 9n € F r (A i). Then 7i U{x, < x ,+ i |t € <*>} is a consistent set of formulas and T\ is universal

4.1 Basic Monomodal Splittings 55

Therefore there exists (g ,< ) ^ T\ such that any point in g has a successor. But then

[T)is a p-morphic image of (<7, < ) € F r (Ai), which is a contradiction. For the converse direction we distinguish two cases: Case 1. A D T = K /Sf [x ]. In this case it is clear that [•]5/-splits A and that A /s* [ • ] = £ is A-persistent. Case 2. There exists n £ such that A 2 K /S NU. Let n be maximal with this property. Define Q := |T]and

Fg := {T £ Frf(A)\F \= A lfi;0T, T is rooted and \F\ < n + 1}.

We check conditions (R I) and (R II) for (G,Fg). Let h £ F r(A) and suppose that a pQ js consistent in h. Then it is easy to show that there exists T £ Fg with

T C h. For (R II) suppose that 7i £ Rfr(A) and T ~ hb for a T € Fg and 6 C h. Take a € A with a D b and let a# denote the n+1 -reflexive iteration of a. By Lemma 4.1.2, {a #} recognizes (G <— hb) in H. H

It follows immediately that 7v(Dn± ) = K /S NU/ S [T]for n £ u>. Clearly F r (A '(D nX )) contains only cycle free frames. Now A '(ü nX) is n-transitive and, by a result of Blok [78], V(/f(DMX )) is locally finite. It follows that any 0 € S K (B U± ) has the fmp. Hence, by Proposition 2.4.27, any 0 £ 5A '(D nX) is a union-5/-split ting of A '(DnX).

Corollary 4.1.4 Any subframe logic 0 with DnX € 0 for ann £ u is strictly Sf-complete and an iterated Sf-splitting of K by finite frames.

Example. Th( [x ] ) = K /Sf (x— / Sf [ • ] .

Definition 4.1.5 For a finite tree T define (T)w C (7 } by

(/ ,r ) € (T )n iff {f ir ) £ (T ) and r C < * U{(z,0)|:r € t} U {(a:,a:)|a: € *}•

Theorem 4.1.6 Let Q £ (T )w , n £ and let A be a R-persistent subframe logic with Fr{ A ) C {/i|AJ g h for fC £ G< U N n}. Then G Sf-splits A and A /S G is R-persistent.

Proof. Define Fg := { G}. Again we check conditions (R I) and (RD). (R I) follows from Corollary 3.2.14. Now suppose that Ft £ Rfr{A ) and Q ~ h*. Take a set {6y|y £ t} which recognizes (T < 0 G ^ hb) in Ft. By induction on dpg(z) we define az £ A for z £ t. For a C h let a* denote the n-reflexive iteration of a.

[ (t>z n y ^ y ^ 0} n n {O M ^ o } ) # if z $zz \ bz O V[{Oay\zsy,yt 0} n f){O b0\zsO} if z z

az is well-defined because Q £ (T)jv. The interesting steps of the proof that {a y|y £ t) recognizes (Q ~ hb) in H are (a ) ay -> Oay = h if y s y and (b ) -> Oao = h if y s 0.(a ) follows from Lemma 4.1.2 by the definition of oy. (b ) Suppose that y s 0 and z £ ay.

56 4 THE LATTICE OF MONOMODAL SUBFRAME LOGICSThen there exists x £ bo with z <\ x. By Corollary 3.2.15 there is a set a C h such that ha has root x and Q ~ ha and { 6y|j/ £ t} recognizes (T <o Q — ha). It is easy to check that for cC / i with hc — Q {a y| y £ t} recognizes (T <o Q — hc) if {&v|t/ £ t] recognizes (T <o Q — hc). Hence x £ ao and z € Oac». "I

Corollary 4.1.7 Suppose that F C (T )m is closed and that n £ lj. Then F defines an iterated Sf -splitting of K/ssN n and A75/N n/S/F is R-persistent.

Fr( ff/5/N n/5/F ) = {h £ Fr\K %h for K £ N n U F }.

Examples.

(i) n ( 0 ) = k /s' 0 / « [ ~ g / « [ j = g

(I I ) b ; defines an iterated ^/-splitting of K andK.Bn = A '(D(n)V B- - ~*Po) = A'/5/Bn-

(III ) For any subframe logic A with A D A / ^ N n for an n £ u the set 1^ defines an iterated 5/-splitting of A and A .Im = A ^ ^ V j - ^ —► ->po) = A / ^ I^ .

(IV ) Define F := { |x—■*!, |«—-x|, | *-*♦ !, |x— x|, |x-— | }.Now T\ = |x—»x| , N i = {7 i } and F C (7j)/v is closed. It follows that F defines an iterated 5/-splitting of K and K/S*F is A-persistent. By the example above I j defines an iterated 5/-splitting of K/S F and A / ^ F / ^ I j is A-persistent. It follows immediately that Kh = K/S F/s If. By a result of N agle & T homason [85] all logics above K5 have the fmp and A 5 is 2-transitive. It follows that all subframe logics above K 5 are iterated S/-splittings of K by finite frames.

4.2 S u b fram e L og ics ab ove K 4 ( I I )

Define Q\ := . Then T2 <0 Q\- Define F j := Q

&2 := I ?e—x I

Qh := x »4 ^

Gz := G4 := <

Cr:=

We assume that the points of the frames Gi are from left to right 0, 1,2, respectively 0, 1. Define T R - := F i U {G2 ■ ■ -Gj)- The following proposition follows immediately.

Proposition 4.2.1 A frame h is transitive if and only if Q %h for Q £ T R “ .

Theorem 4.2.2 For n £ u A „ := K/S*T*/s*F i/stQ2/ss . . . /5/Q7 is well-defined and A „ = K (S f2 (G)\g £ T R " U {7 ;* }) = K A f^T*.

57Proo f. Let n 6 « . Tn* and 72 are cycle free and therefore 5/-split K and K/S*T * Is*%. is Ä-persistent. Define m := m ax{n ,2 }. Then K/S T*/s Tz 2 K/s*N m- Now F i C and F i is closed. It follows that F i defines an iterated 5/-splitting of K/S*T* and that 0 i := K/sfT */5 F i is Ä-persistent. We prove by induction :

(I ) For 1 < j < 7 : Gj+i Sf-splits 0 j := Oi/s*Gz /SfGj and ©j+i = ®j/S*Gj+1 *s Ä-persistent.

4.2 Subframe Logics above K4 (77)

For 1 < j < 4 define F$.+1 := {t/j+i}. We check conditions (R I) and (R II) for Qj and (Gj+i, Fg +l). For b C h€ Fr let b* denote the m-reflexive iteration of b.

j = 1. (R I) It is easy to show for h € F r (0 i ) : Gz Q h if □ (2)V $2 A p0 is consistent in h. (R II) Suppose that Gz ^ fit for a 6 C fi and € Rfr(Qi). There exists { 60, 6i } which recognizes (T] <0 Gz - fit) in W. Define aj := 61 n O60 and a0 := 60 H Oc^. It is easy to check that {a 0,a i } recognizes (Gz - fit) in H. (One only has to use Tz £ fi).

j = 2. (R I) Suppose that fi € F r (0 i ) and fi,/?,z A po- Then z € /?(po) andthere exist z i,p i,p 2 with *1 € /?(Po),Pi,l/2 € /?(pi) and z <1 pi <1 xt <1 p2 and pi-^ p2,P15I Pi. Case 1. z = X]. Then fi{y,,Xl} — G3 because Gz % fi- Case 2. z / xi. Thenp2 < z i and again fi{Xl>V2} — Gs- (R II) Suppose that G3 — fit and H € Rfr(Q2). Let{&o,6i } recognize (7i <0 £3 ^ fit) in H. Define a\ := 61 fl O60 and Oo := (60 0 Oao)*. The problematic step of the proof that {oo ,O i} recognizes (G3 — fit) is ai —► Oao = fi. Suppose that z € o j, z £ Oao. Then there exists p € 60 — ao, z € 61 with z <1 p <1 z, z ^ z, x<f\ z, zf\ z. But then z<] y and y<i y because Gz (2 fi- It follows that y 6 Oo! a contradiction.

j = 3 . (R I) Suppose that fi € Fr(0 3) and fi,/?,z (= A po- Then there existPi,P2 € ^(pi),zi € /?(po) and zi € ß(pz) with pi < zl5zi < p2,zi < *i and y ifiz i. HenceZi < zi and zi <1 z\, zj < Zi. From Zi < Zi < P2 and Zi 4 yz follows yz < zi and yi < y\. Hence fi{Xl>V2>Xl} — Ga- (RH) Suppose that Ga — fit with H 6 Ä/r(03) and that T is the tree with T <0 Ga- Let {60,61,62} recognize (T <0 Go — fib) in H. Define a, := (6,nO60)* for * = 1,2 and ao := (6of)Oai nOa2)*. The problematic step of the proof that {ao, ai, 02} recognizes (Ga — fit) is a; —► Oao = fi for * = 1,2. We prove this for * = 1. Suppose that z € ai,x £ Oao. Then there exist xi € 61, y € 60 ~ <*o and z € 62 with z< p,p< Zi,p< z and z-$ z,zi^j z,z-f) z,z-^ zj. But then z < p and p < p,z < z. Now z < p < zi and therefore Zi <1 p and Zj <] Zj. Hence p € ao, which is a contradiction.

j = 4. For this step (R II) is false. We will show (R I) for Fgs := {Gs} and then prove that for H € Ä/r(0 4) with Gs — fib there exists a generated subframe G o f a subframe of H such that b C g and (Gs — 9b) is recognizable in G- That Gs S f -splits 0 4 and Ä-persistence of 05 follow immediately. (R I) It is easy to check that Gs C fi if d (2)V(;s A p0 is consistent in fi. Now suppose that Gs — fib via an isomorphism p : Gs — ► fib with p(0) = z € fi. Let { 60,51,62} recognize (72 <0 Gs — hb) in H. Define Tf\ := 1fboubiub2 and let G denote the subframe of Hi generated by z. Define c, := 6; n fix for t = 0,1,2. {co,ci,C2} recognizes (72 <0 Gs - 9b) in G- Define a2 := (c2 fl O ci)*,a i := (ci n Oa2)# and a0 := Co n Oai. We prove that {a 0,a i,a 2} recognizes (Gs - 9b) in G- The problematic step is a2 —*• Oai = g.

58 4 THE LATTICE OF MONOMODAL SUBFRAME LOGICSSuppose that y € Q-i- Let n g w b e minimal with y 6 Tr*(x ). Co n Oc2 = 0 and Co = {x }

and therefore n > 2. There exists a path x = xq < x\ < .. . < x„ = y with x,fl Xj for j > t + 1 . It follows by induction that x1+i <3x, for i > 0. From this follows, by Q2, S3 2 SN that x,' is reflexive for i > 0. Again it follows by induction that x, 0 because Qa £ 9• Hence y <i x\. Now xj 6 61 n 062- Therefore there exists z 6 62 with xj < z. It follows that z <J i j and z <\ z. Hence xi € cti and y 6 Oaj.

The step j = 5 is proved anlogously.

j = 6. Define Fq1 := {£ 7}. (R I) follows from Corollary 3.2.14. (R II) Suppose that Q7 ~ lib with H e Rfr(0e). Let { 60,6i , 62} recognize (T2 <0 Ö7 — H ) in H. Define a2 := (62 n Obi H O60)*, ai := (61 H Oa2 fl O6o)* and ao := (bo H Octi)*. It is easy to check that { 00, 01,02} recognizes (Q7 ~ kb) in Ti.

(I ) is proved. From (I ) and Prop. 4.2.1 follows An = K 4/S*T ’ . The axiomatization of A„ follows immediately. H

4.3 A Chain of incomplete Subframe Logics

For the following construction we use ordinal arithmetic and the notion of intervals in the usual way. For two ordinals Q i,Q2 let [01,02] denote the set {7 € ord|aj < 7 < 02}. Define (h, < ) by h := uu> + 1 and

< := {(un + mi,um + m2)|n,mi,m2 €w,mi > m2}U {(u>a,u;a)|a € u;+ l,a ^ 0} U {(u>(n+ l),um + m)|n, m € u;}.

Now define 7i := (h, <, A), where A := [0]* denotes the closure of 0 C h with respect to finite intersection, complement and □. Define Into := u> + 1 = [0,u;] and for n > 0 define Intn := [um + l,u>(n+ 1)]. Then h = |J{/»tn|n € u;} U {ww}. Define a set C by c 6 C iffcCfc and there is an m G u with c C [J{/ntn|n > m} U {wu>} and for m < n either c fl Intm is finite and u>(m + 1) £ c or Intm — c is finite and u>(m + 1) 6 c. Define B := {c|c € C or — c € C}.

Claim 1. A = B.Proof. B C A. The following equations are easily checked:

1. Into = OD0UD0.

2. 7ntn+1 = —Intn 0 0/nt„.

3 . For m € u> : m = □m+10.

4 . For m € u;: {m} = m + 1 D — m.

5. Forn ,m e«: {w(n + 1) + m + 1} = - □ ”*/«<„ fl □TO+1 In tn.

4.3 A Chain of incomplete Subframe Logics 59It follows that In tn £ A for n £ u> and that { a } £ A if a = 0 or a is a successor ordinal. From this follows immediately that B C A. A C B: It is clear that B is closed under finite intersection and complement. We show that B is closed under □. For b C h define mmn(6) € ww if IntnC\b / 0 by minn(b) := min(br\Intn). Then Obnlnt0 = [0, mm0(6)+l] if b n Int0 / 0 and D6 n Into = { 0} else. We have for n > 0

f [um + 1, minn(b) +1] if b H In tn 0, un £ b □6 n In tn = < 0 if un £ 6

{um + 1} if b D In tn = 0 and urn € bIt follows immediately that B is closed under □.

For any ordinal a with a + 1 < u;u> + 1 define 7ia = (ha, < a, Aa) by

hQ := a + 1, < Q := < n h* and Aa := {a n (a + l)|a £ A).

Let AQ := Th(Ha) and A := A ^ . Any 7fa is a generated subframe of andAa = [0]/ia- Wu,3+i looks as follows:

u>3 + 1 u>2 u> 1 0

The logic 6'.3 = (7./i will play a fundamental role in our investigation of incomplete subframe logics. The rooted G.3-frames are the irreflexive linear orders without infinite ascending chains. The set of finite rooted G.3-frames is, up to isomorphism, the set {T n*|n G u>}.

Lem m a 4.3.1 For all a u>:

1 . Hq is descriptive.

2 . H i is subdirect irreducible if a = 0 or a a successor ordinal.

3. Aa = CiiAßlß < a ) if a is a limit ordinal.

4- A „ is a subframe logic.

5. Fr(Aa) = Fr{G.3) if a ^ uju and a > u>.

P roo f. 1. It is clear that Ha is refined. Suppose that U is an ultrafilter in W+. If U contains a finite set then f| U ^ 0. Now suppose that U contains no finite set. Then a = ujß + m for a ß ^ 0 and ß £ u> + 1 and m £ u. But then it is easy to check that there exists ß\ < ß with uß\ £ a for any a £ U. Hence f| (/ / 0. 2. I f a = 0 then {0 } is an opremum of Hq . If a = ß + 1 then (a + 1) - {ß} is a opremum of Ti+. 3. Suppose that a is a limit ordinal and 7 is a valuation of Ha with 7(<£) / 0. Let 7(<j>) = a £ Aa. Then, by Claim I, there is a ß < a with ßr\a ^ 0. Define a valuation 71 on Hp by 71 (p) := -f(p)r\hp. Then 71 {<£) # 0. 4. Let a £ Aa. We show that (Ha)a is a Aa-frame. Then it is shown that Aa is complete with respect to a class of frames closed under subframes and therefore

60 4 THE LATTICE OF MONOMODAL SUBFRAME LOGICSis a subframe logic. Consider a rooted generated subframe T of {Ha)a- It is obvious that there is a ß < a with f ~ Hp. But then F (= Aa. 5. Fr(G .3) C F r (Aa) is clear. Now

let a = urn + n. Then Aa is m + 1-transitive. Q := [• ] £ Fr(A a) because Vg ;po isnot consistent in Ha. Now suppose that there is an intransitive frame T € F r (A a ). Then 7ä C T because Q Fr(A a). But then o (2)V t2 ; po is consistent in Ha. By Corollary 3.2.14 72 C ha. Since this is false, Fr(Aa) contains only transitive frames. Clearly Ha \= h- These observations together imply Fr(A a) C Fr(6'.3).H

Corollary 4.3.2 For u + l < a < uu the Aa arc different incomplete subframe logics.

Corollary 4.3.3 For n > 1 we have FnsK.Trn+i(K .Trn) D { O A\Trn|m > 1}. Hence K .Trn has Sf -degree of incompleteness > H0 above K .Trn+l.

Proo f. Define 0 := Awn+m H K.Trn.It follows immediately that € Rfr(O) and Wwn+m ^ Rfr(K .Trn). ThereforeK.Trn+1 C 0 C K.Trn. We show F r (0 ) = Fr(K .T rn). 0 is a subframe logic complete with respect to {Tfum+m} U F r(K .T rn). Hence, by Corollary 2.4.13 and example (III) on page 51, F r (0 ) = Fr{K .Trn) if Tr^ n Fr(7 fum+m) = 0. But, by Lemma 4.3.1 (5), this is the case.

C orollary 4.3.4 For n > .1 the logic K .Trn is not strictly Sf-complete.

So far we have only examples for ^/-splittings by finite frames. Splittings of the lattice SA provide nice examples for splittings and ^/-splittings by infinite frames.

Lem m a 4.3.5 Let Q = {g,<, A) be a descriptive frame and A generated by 0. Let { x } € A and let Qx denote the subframe generated by x in Q. Then Qx splits A := Th{G). If A is n-transitive for an n £ u then A/Gx — Th(Gb) with b := □ In)(p — {x } ) .

P roo f. It is clear that I : Fr^Q ) — ► G+ ,4> </>[0], is an isomorphism. gx— { x } is anopremum of G* and therefore Gx is subdirectly irreducible. Let / : G+ — *• Gx ,a »-*• af)gx, be the canonical homomorphism. Define pr := / o I : Fr\(Q) — ► Gx and let A be a corresponding diagram of Gx and ij> € C with pr(V>) = {x } . Suppose that i/> is consistent in A € V (A ). For <j> € A , k 6 u we have A I- i> —*• □ k<f> because t/> and 4> are constant formulas. It follows that D ^ A ; V> is consistent in A. Hence, by the Splitting Theorem, Gx splits A and A/Gx = A(->i/)). Now suppose that A is n-transitive. Gy, is a generated subframe of G- It is clear that Gb € Rfr(A(->i/>)). Take a <f> # A(->if)). Then o ln)-iif> —* <f> £ A and there exists a valuation ß of G and y € g with G,ß,y |= o (n)-nf>A-«t>. But then y € Gb- H

Corollary 4.3.6 Let a be a successor ordinal, a = at\ + 1. Then(1) Ha splits and Sf-splits A ^ .(2) For uu > ß > a : Ha splits Aß and Ap/Ha = Aa, .

Proof. A simple consequence of Lemma 4.3.1 and Lemma 4.3.5.H

4.4 Subframe Logics above K4 (HI) 61

Corollary 4.3.7 For all a < uu : SAa = £Aa = ( { Aß\ß < a } U {£ } , C) ~ (a + 1, 3).

Proof. Consider a 0 £ £Aa with 0 ^ £, Ao- Let ß be minimal with 0 3 A/*. If ß is a successor ordinal, ß = ß\ + 1, then A^ splits £Aa and £Aa/Aß = A ^ . Thus, 0 = A/j. If ß is a limit ordinal then Aß = f^A^J/?! < ß ). Now any A ^+ i with ß\ < a splits £Aa and £Aa/Aß1+i = Aßl. Hence 0 C Apx for ßi < ß and therefore 0 = Aß. H

4.4 Subfram e Logics above K 4 ( I I I )

Theorem 4.4.1 Suppose that F is a set of finite, transitive and rooted frames and that n > 2. Then the following conditions are equivalent:(1) A'4/5 F is strictly Sf -complete.(2) K.Trn/ sf (T R ” U F ) = A4/5/F.(3) A\7>„/^(TR “ U F ) has fmp.(4) K.Trn/ sf(T Il~ U F ) is complete.(5) 3m eu> : T * e F.(6) Ä '4/^F is an iterated Sf-splitting of K by finite frames.

Proof. It follows from Prop. 4.2.1 that F r (K 4) = Fr(K .Trn/ sf T R ). Therefore

(I) Fr(K 4 /s Y) = Fr(K .Trn/ sI (T R ” U F ))

for any set of finite rooted transitive frames F. (1) => (2) follows immediately from this observation. The implication from (2) to (3) follows from the fact that any subframe logic above K4 has fmp. (3) => (4) is trivial. (4) => (5). Suppose that T*x F for all m £ u. Then TC+i £ Rfr(K.Trn/ sf(T R ~ U F )) because Fr{K w+1 ) = Fr(G.Z). But TC+i £ Rfr(K4) and therefore, by (I), K.Trn/ sf (T R ” U F ) is not complete. (5) =$► (6). Suppose that £ F. By Theorem 4.2.2 K4/S T^ is an iterated 5/-splitting of K by finite frames. Hence K4/S*Y is an iterated 5/-splitting of K by finite frames. (6) =$► (1). Strict Sf-completeness of Ä '4/^F follows from completeness of K4/S*Y. H

CoroUary 4.4.2 For A £ SK4 the following conditions are equivalent:(1) A g G.3.(2) A is an iterated Sf-splitting of K by finite frames.(3) A is a union Sf-splitting of K.Tr2 by finite frames.(4) A is strictly Sf-complete above K .(5) A has strict Sf-fmp above K.(6) A is strictly Sf-complete above K.Tr2.(7) A has strict Sf-fmp above K.Tr2.

62 4 THE LATTICE OF MONOMODAL SUBFRAME LOGICSProof. We know that all subframe logics above K 4 are union-5/-splitting of K4. Let A = KA/^F for a set of finite and rooted transitive frames F. It is easy to show that A %. G .3 iff there is an m € w with T*x € F. Now all the equivalences follow immediately from Theorem 4.4.1. H

Clearly we can replace by Ha, u> < a < uu, in the proof of Theorem 4.4.1. Hencewe get for a subframe logic A = KA/S F C G.3, F a set of finite rooted K 4-frames:

For n > 2 : FnSK.Trn(A) = [K .T rJ s* (TR~ U F ), A] D {A H Aa|u> < a < wn}.

It follows that A has Sf-degree of incompleteness > K0 above K.Trn for n > 2. The question arises whether there exist subframe logics above K4 with Sf - degree of incompleteness 2k°.

Theorem 4.4.3 Let K4 C A C 6', A a subframe logic. Then A has Sf-degree of incompleteness 2Ko above K.Tr2.

Proof. Consider the frame Hu, = (ho,, < w, Aw), defined on page 58, and a finite (7-frame G — {g, < s) with root z. Assume that g n hw = 0. Then define Gu> — (du, by

9u, := K U g.

< 9u := U < ff U {(z ,w )}.

Bu, := {6 C gjßa € AJ^c C g : i = o (Jc }.

It is readily checked that

Gu, |= K.Tr2 and Fr(Sf(Gu,)) Q Fr(G.3) U Fr(Sf(G )) Q Fr(G).

A simple variant of the construction in Zakbaryaschev [92a] shows that there exists an infinite set N of mutually <sf incomparable finite rooted G-frames. Now consider a subframe logic A with K4 C A C G . Define for M C N :

A m : = a n Th{{Sf{Gu,)\G € M }) .

Clearly, Am (= K.Tt2 and F r(A M ) = F r(A ). We show that Am j # Am 2 for M i % M 2.

Take a G € M i — M 2- Define <f> := A pz) —► (Op —*• DDp), where z is the root ofG and p £ var(Vg). It is clear that -><f> is consistent in Gu and therefore Am j |£ 4>- Butit follows immediately from Th{Sf{T )) % Th(G) for all T € M 2 that Am , b <£. H

*2

An analogous construction with Hw replaced by Hum shows

63

Corollary 4.4.4 For n > 1 the logic K.Trn has S f -degree of incompleteness 2K° above K.Trn+1.

The question arises whether all subframe logic above K4 and weaker than G .3 have .57- degree of incompleteness 2**°. We conjecture that this is not the case. The proof delivered above shows that all subframe logics A above A 4 and weaker than G .3 with |«SA| = 2 ° have Sf-degree of incompleteness 2K°. We conjecture that this also holds for 2K° replaced by N0. In this case it would follow that the .5/-degree of incompleteness of G .3 is N0.

Let A = A'4/s/F £ G.3, where F is a set of finite rooted K 4-frames. Then it follows from Proposition 2.4.24 that a logic 0 is a lower cover of A in S A f if and only if there is a frame T £ ms/(T R ” U F ) with 0 = A fl T h (S f(F )). Hence for a finitely axiomatizable subframe logic A above A' 4 and not weaker than G .3 the lower covers of A in S A f are easily determined. We will now describe the lower covers (in S A f) of subframe logics between G and 6\3.

4.4 Subframe Logics above K4 (HI)

Theorem 4 .4.5 Let A = G /s*F C G.3, where F is a set of finite and rooted G-frames. Then the lower covers of A in S A f are

( I ) A n T h (S f(F )) for T £ mSf( F).

(^ A n r/ H H ).

(3) A Cl Th( X- *X )•

(4) A n Th(Hw+i).

Hence if A is finitely axiomatizable then A has finitely many lower covers in SAf and ij A is not finitely axiomatizable then A has No lower covers in SAf.

Before starting the proof let us note some remarks and consequences. We know from the results of Blok that a logic which is not strictly complete in AT has 2K° lower covers in Af. Theorem 4.4.5 tells us that in S A f the situation is different. There exist subframe logics with 5/-degree of incompleteness 2K° but with only finitely many lower covers in S A f. For instance, G has exactly 3 lower covers in S A f but has 5/-degree of incompleteness 2K°. Even if Kripke-semantics is not strong enough to seperate a logic from its lower covers in S A f , the lattice itself can be rather thin. We believe that the same is true for all subframe logics above K4 but have not been able to prove this. A nice and easily checked consequence is that we have for a finitely axiomatizable subframe logic A above G that for <t> £ C it is decidable whether A = K(<l>Sf). We start the proof of Theorem 4.4.5 with a lemma.

Lem m a 4.4.6 Suppose that A is a subframe logic with K.Trn % A for all n £ u). Then 0 € F r (A).

64 4 THE LATTICE OF MONOMODAL SUBFRAME LOGICSProof. Let A be a subframe logic with A 2 K-Trn for all n € w. Hence for n 6 « there is a W G Rfr(A ) and a valuation ß such that 7f,/?,x f= □ (n+1)v . j r- A po, where

U-f 1T r “ +1 = A.‘|+1. Define b := U{^(Py)l2/ € K.n+1} and 'y(p) := b. It is readily checked that

f= m(n)(p —*• <>p) A p. Hence □ (")(p —► Op);p is consistent with A and therefore

0 € F r(A ). H

Proof of Theorem 4.4.5. Let A be a subframe logic with G C A C G. 3. It follows from Proposition 2.4.24 and Corollary 4.3.7 that the logics defined in the theorem are lower covers of A in SAf. Now suppose that 0 is a lower cover of A in SAT.

Case 1. 0 D K4. Then, by Proposition 2.4.24 and the fact that G = K4/Sf |~*~| ,0 is one of the logics defined in (1) and (2).

Case 2. 0 2 F4 and there is a H = (h, < , A) £ R fr(0 ) such that Q C h for a frame G € F := {T 2, G2, 0&,. . . , £12} , where

IIX , g2 = [x~*|, Gs = rx V ' “ A B A

,{? !! = > Ö12 =

It follows immediately from the proof of Theorem 4.2.2 that {72, G2} defines an iterated ^/-splitting of K such that K/S T2/S G2 is Ä-persistent. Now F — {G2} Q (T2)r and therefore it follows from Corollary 3.2.18 that F — {G2} defines an iterated S/-splitting of K ls*T2/slG2 such that K/Sf F is Ä-persistent. Hence, if G C h for an H £ Rfr(Q) and G € F, then there is an T £ F with T £ F r (0 ). Now, if T — T2 € F r (0 ) then © = A n Th(T2) and if T £ F - {T 2} then 0 = A D Th{ [T ] ).

Case 3. Else. If [• ]€ F r (0 ) then © = An77»( [• ] ) . Now assume that |T] F r (0 ). By Lemma 4.4.6, there is a minimal n £ u> with 0 ^ T rn+\. We have n > 1. There is a refined 0-frame % = (h ,< ,A ) with H ^ □ (n+,)V^y- Apo. Hence there are sets bo,...,bn € A with bi ^ 0 for i < n and 6,- H 6j = 0 for * ^ j and 6,- —► 06t+i = h, 6,- 0 Obj = 0 for j > i + 1. Take a sequence x<> < £1 <1 . . . < xn with xt- £ 6«. It follows from [• ] £ F r (0 ) and 0 ^ T rn+i that

(I ) For all reflexive x £ h and b £ A with x £ b and m £ ur.Hb,x (= <>(n+1)(aTO+1x A iD m±).

Hence, by minimality of n, we may assume that bn n 06n = 0. It follows from G % h for G € F that for all x £ &n_2, z £ 6n_ i with x < 2 we have that z is reflexive. Hence xn_ 2 < xn_ i <J xn and xn_ i is reflexive, xn is irreflexive and xn_2-$4 xn. Now let m £ u>. Then Win_ ,,x n_ 1 |= 0^n+1^(Dm+1± A -iDm± ). It follows again by minimality of « that there is a set of irreflexive points {z 0 < < . . . < zm+ i } C 6n_ i and a reflexive point yin &n_ i with xn_2 < y < zq and x „_2 ) zt , z ^ y for t < m + 1. Hence there exist sets co,. . . , Cro+3 6 A with Co # 0 and

4.4 Subframe Logics above K4 (II1) 65

( i ) co -► O ci = h, co n Oci = 0 for i ^ 1 ,

(ii) ct H Ocj = 0 for i > 2 and 0 < j < z,

(iii) ct Ocj = h for 1 < i < j.

W e show that there are x 6 Co, y € c\ with x < y and x. Suppose that this is not the

case. Then define d\ := OcoHci. It is readily shown that (coU d i) —► O (co U d i) = h. Hence

Q e F r (0 ) , which is a contradiction. It follows that there are sets a o ,.. . ,a m+ 3 6 A with

the properties (i),(ii ),(ii i ) and a, fl Oao = 0 for i < m + 3. Define a := ao U ... U am+ 3 . Then we have for x 6 flo-

Ha,x (= DOT A -«Dl A f$nk(0(nm+l± A -.DmJ_) V j_))|l < k < n + 1>

It follows immediately from Q % h for Q 6 F that

r := DOT A -.D-L; {A(D*(0(Dm+1± A -.Dm±) V Dm+1 J_)|l < k < n + l>|m € u}

is consistent with 0 . Now take a descriptive 0 -fram e H with root x such that ?{,x (= T. Define p : H — ► H^+i by

p-1({u; + 1}) := {z 6 h\7i,z (= DOT A -iOJ_} p-'iim}) := {z € h\7i,z (= □Tn+1 j_ a -.Dm±}

:= h - U{P”1( {a))la € w,a = u; + 1}.Using the assumption that Q % h for Q 6 F it is readily checked that p is a surjective

p-morphism. Hence 0 = An Th(H„+$.A

66 4 THE LATTICE OF MONOMODAL SUBFRAME LOGICS4.5 Simple incomplete and not strictly 5/-complete Logics

In this section we use some variants of the frames Ha to give examples of incomplete subframe logics with simple axioms and of Ä-persistent, not strictly 5/-complete, subframe logics. For expositional reasons we will not formulate the theorems as general as possible but in a way that makes it possible to transfer them easily to other classes of logics. We hope to convince the reader that, with the tools we have at hand now, it is more or less a child game to produce incomplete subframe logics. First we need some notations for isomorphic copies of frames.

1. F or i e g,y&g, let (s,<)[s//x] := ( (p - {x } )u {y } , <U {(y , z)|x< z}U {(z, J/)|(z<lx}).

2. For to € w define (g,<l) x { to} := {g x { to} , { ( ( x , to), (j/,m))|x <1 3/}}.

For a finite tree T with root 0 define for x € t, x ^ 0 the structure Tr(x) := (t, < U{(x, x)}). That is, we simply define a point x € t to be reflexive. Let Tx denote the subframe generated by x in T and define T-x := (t~x, < f l (t— fx)2), where t_x := t — tx. Define T (—x) := (t_x U {x }, < n(t_x U { x } ) 2). Now define

(go,<go) := (Tx x {0 })[0/ (x ,0 )].

For to ^ 0 : := (% x { to} ) [u>to + l/(m ,x)].

For to ^ 0 : ■- ( T ( - x ) x { to} ) [wto/(to, x )].

Remember the definition of = {/ w , < Mm,Aum) on page 58.For to € w, to ^ 0, we define Hm(T, x ) = (hm(x ), < x, Ax) by

hm(x) := hum, U f m U JJ{<7n|n < ”*}

< x := <W U < / m U (J {< sJn < to} U {(y,z)|y € (In tn - gn),z e gn,n < to}

Ax {c C fim(x)|3a € A^Sb C (/m - {u>m}) U (J{s„|n < to} : c = a U 6}

7iz[T, x) looks as follows:

We omit the proof that Hm(T ,x ) is a refined frame. It is similar to the proof on page 59. For to € t*; let (m < Tx) denote the subframe generated by { to} in H i(T ,x ).

(to < % ) =m

%

1 0

4.5 Simple incomplete and not strictly Sf -complete Logics

L em m a 4.5.1 Let T be a tree, x £ t and x ^ 0. Then

67

(1 ) For n > 0 : F r f{S f {H n{T , * ) ) ) = F rf (S f ({ (m < % )\m € u,} U {T _ x } ) ) .

(2 ) For m > dp(T ) : □ (”l)VTr(l) A p0 consistent in H m(T ,x ).

(3 ) For m > dp(T ) : ( □ ( 2" ‘>VTr(i) A p0) } q is not consistent in Ttm{T ,x ).

P ro o f. (1) F tj{Sf(TLn{T ,x ) ) ) D F rf (S f({{7n < %)\m £ u } U {7 1 x} ) ) is clear. Suppose that Q £ F r/ (S/ (? fn(T , x ) ) ) is rooted. Then there exists a set b E j4x such that £ is a

p-morphic image of (H n(T , x ) )b via a p-morphism p. We may assume that (H n{T , x ) )b is rooted, b contains no limit ordinal: Suppose that ujvi E b and p (ujvi) = z £ g. Then, since umi < um, we have z < g z and therefore p_ 1 (z ) C Op~l (z ), which is clearly impossible. Hence H n(T , x )b ~ (m < T x) or (H n(T , x ) )b ~ 7LX. (2 ) Define a valuation on 7im(T , x ) by

ß(px) := / i^ , /?(py) := { (y ,rn ) } for y € t - * and ß{py) := { ( y ,n ) l » < for V ^ t x,y ^ x . It is easy to check that H m(T , x ),ß ,(0 , to) |= □ ( ’K)V 'rr(l) A po- (3 ) follows immediately from 277i-transitivity of H m(T ,x ), Corollary 2.4.13 and (1).H

C o ro lla ry 4.5.2 Suppose that T is a tree with x E t , x / 0 . Then (1 ) Frf(Sf(TCn(T ,x ) ) ) contains only cycle free frames for n > 0. (2 ) If x is not a successor of 0 or 0 has at least two successors, then T £ F r j (S f (H n(T , x ) ) ) .

T h eo rem 4.5.3 Suppose that T is a tree with root 0 and that x £ t,x ^ 0, is not a successor of 0 or 0 has at least two successors. Then the following holds for n > m := 2dp(T):

(1 ) 0 n := (A '/5^ T ) (S fn(7^(x) ) ) is an incomplete subframe logic withFr{en) = {h£Fr\T,TrM %h}.

(2 ) ®mi ^ ® mj for i ^ j with i j > 1 .

(3 ) Tr(x) does not Sf-split K / S T.

P ro o f. Define k := dp(T ). ( 1 ) We have, by Proposition 3.2.13,

F r (Q n) = F r (K (ü<fc>VT < - -^po)) = {h € Fr\T,Tr(s) % h } for n > k.r(x)

By Corollary 4.5.2 and Lemma 4.5.1, H k (T ,x ) 6 Rfr(Q n) and, by Proposition 3.2.13, H k(T ,x ) C lW V ys -»po* ©n is incomplete because there exists a complete logic

different from 0 n with the same Kripke frames.(2 ) By Lemma 4.5.1, Hm ,i(T,x) <= Ä /r(0 ro(l+1) ) and Wro(l+ i ) ( 7 » £ R fr (e m{i+l)) for i > 1 . Hence 0 mt ^ 0 m(,+1) and, by 0 mt C Q mj for t > j , 0 m, ^ 0 mj for i ^ j . (3 ) follows immediately from ( 2 ) and the Splitting Theorem.H

68 4 THE LATTICE OF MONOMODAL SUBFRAME LOGICSHence with any tree T with \T\ > 3 there is associated a quite natural descending chain of finitely axiomatizable incomplete subframe logics with a basic elementary class of Kripke frames.

Exam ple. The perhaps simplest incomplete subframe logic is given by

* ( S f l ( I i ) , S f 2(G i)), where I , and G\

The condition x 0 and x not a successor of 0 or 0 has at least two successors cannot be omitted. Consider the following two examples. Define

Then, by Corollary 3.2.18, {l\,Go} defines an iterated 5/-splitting of K and K js X\/s Go is Ä-persistent with Fr{K/s l\/^Go) = {h 6 Fr]fL\,Go 5? h}.

Remember that 7i = | x—-x| and define T := | x—— 1.

Then, by Corollary 4.1.7, {7 j , F } defines an iterated 5/-splitting of K and K !s T\js T is fi-persistent with Fr(K / s T\/s Jr) = {h € F r|7 i,F % h}.

Theorem 4.5.4 Let A be an R-persistent subframe logic and suppose that there exist a tree T and x € t such that T ,T r(x) & F r (A). Further suppose that (m <J Tx) € F r (A ) for all m € u> and that 71* € F r (A ). Then A is not strictly Sf -complete.

P roo f. Let k := dp(T) and define A j := Th(Sf{Hk(T, x )))n A . F r (A i) = F r (A ): Suppose that there exists h 6 F r (A i) such that h £ F r(A ). Then there exists a finite rooted subframe g of h with g £ F r (A ). A i is a subframe logic and therefore g 6 F r (A i). Then g € F rf(S f{H k (T ,x ))) and, by Lemma 4.5.1, g € Frj(S f{{{m < %)\m 6 w } U {7 1 * })). But then g € F r (A ), which is a contradiction. A i ^ A because, by Corollary 3.2.14, A h —► ->po but, by Lemma 4.5.1, H k(T,x) ^ -* ->po. H

Corollary 4.5.5 Suppose that T is a tree and that T — (/, r ) with 7 < 0 7 and

r c po) is not strictly Sf-complete.

P roo f. By Lemma 4.5.1, A '(ü ^ p^ )V ^ -< —► ->po) satisfies the conditions of Theorem4.5.4.H

4.6 A note on Neighbourhood Semantics 69

Example. K.In is not strictly ^/-complete because In satisfies the conditions of Corollary 4.5.5.

Notice that for any T satisfying the conditions of Corollary 4.5.5 we have that the logic (Ar/5 N 11)(D ^ p ^V^-< —► -ipo) is strictly 5/-complete because T - C (T)j\j. This observation shows once more the fundamental role of 5/-splittings by N n.

4.6 A note on N e igh bou rh ood Sem antics

This paper deals with algebraic and Kripke semantics for modal logic. A well known alternative approach is the neighbourhood semantics. It is known that there exist normal modal logics which are complete with respect to neighbourhood semantics but incomplete with respect to Kripke semantics (consult Gerson [76].) But all known examples require the construction of rather difficult generalized frames. The logic Th(7tu,+i) provides a rather simple subframe logic with these properties. For neighbourhood semantics we use the notation from Gerson [76]. A neighboorhood frame is a pair (g, N ) such that N is a function from g to sets of subsets of g. The theory of a neighboorhood frame (g ,N ) is denoted by Th({g, N )).

Theorem 4.6.1 Th(7fu;+i) is complete with respect to neighbourhood semantics.

Proof. Define g := u U {u> + 1}. For n € let N (n ) := {a C g\a D {0 ,.. .n — 1 }} and let N (u + 1 ) be a nonprinciple ultrafilter on g. It is readily checked that Th(Hu,+i) = T h ((g ,N )). H

The notion of strict 5/-completeness with respect to neighbourhood semantics is defined in the obvious way. By Theorem 4.4.5, we get the following

Corollary 4.6.2 All subframe logics above G are strictly Sf-complete with respect to neighbourhood semantics.

4.7 Subfram e spectra

If 0 is an iterated 5/-splitting of A by finite frames, then we know from Proposition 2.4.30 how to determine the subframe-spectrum of 0 above A.If 0 = A/5 F then Sps\(0 ) = {0 i € £A|0i C 0 and g £ F r (0 j ) for g € ^ 5/ (F )}. If ms/(F ) defines an iterated 5/-splittung of A then Sj>5a (0 ) = [A/ms/(F), 0 ]. Hence for all 0 D K4 the subframe-spectrum of 0 in €K4 is an interval. Let us note some examples:

Define 54.2 := 54 (ODp—► DOp). Then 5 p$$4( 54 .3) = [54.2,54.3], because

, (Rautenberg [79]) and 54.3 =54.2 = 54/

70 4 THE LATTICE OF MONOMODAL SUBFRAME LOGICSDefine 54.1 := 54(DOp —*• ODp). Then 5p,s54(54.1) = [54.1, Grz), because

54.1 = 54/| » —• |, (Rautenberg [79]) and Grz = S4/Sf | .

Spsk*(G ) = [Ä'4(ü-L V O D ±),G ], because K4(D± V ODJ.) = K4/ [*]and G = K4/Sf [T ].

For a finite, cycle free and rooted frame g we have Spsk (K/SJg) = [H/g, K/S}g). Especially S p s k (T ) := [F (O T ),T ].

Spsi<(Th( 0 ) ) = {A| 0 € Fr(A),|x—*x], [*~]£ Fr(A)}.

SPs k (TH H )) = {A| H € F r(A ), 0 , 0 E 3 , 0 = 3 t F 'r(A )>-

SPs k {Gtz ) = {A C Grz\ 0 , 0 3 , , 1» - -| t F r (A )}.Hence A'(DC>p —♦ ODp) y&= Grz.

Now suppose that 0 is not an iterated 5/-splitting by finite frames. In this case it is in general not so easy to prove that 0 j 0. But if we know the lower covers of 0 in SMy then again straightforward reasoning is possible. As an example let us show that F(DJ. V ODJ.) t 5 = G. This follows if 02 ^ a -L V OCU_ for all lower covers 02 of G in SAT. But this follows immediately from Theorem 4.4.5.

715 S/-completeness in Polymodal Logic

We start our investigation of ^/-completeness in polymodaJ logic with some negative results.

Proposition 5.0.1 Let A ;,i < rt, be monomodal subframe logics with A, ^ £. Then all A,• are strictly Sf-complete if 0 (A ,| i < n) is strictly Sf-complete.

Proo f. It is shown in Kracht k WoiTER [91] that Ai = 0 (A ,| i < n )f l£ i . Now suppose that A] is not strictly ^/-complete. Then there exists a monomodal subframe logic 0 C Ai with F r (0 ) = F r (A i). It follows that F r (0 & 0(A,|2 0(Aj|2 < i < « ) C 0 (A ,| i < n). Hence 0 (A ,| i < n) is not strictly Sf-complete. H

With this easy observation at hand we can extend the negative results concerning strict S/-c.ompletenes from monomodal logic to polymodal logic. The converse direction is not valid. Before giving a counterexample we define some bimodal frames which we use to deduce some more basic counterexamples. Define

h := u + 2 and <1 := {(i,/)|i,/ < u;,i > / } U {(u>,u;)}.

A := {c C h\u € c and c cofinite or u £ c and c finite }.

Now let Ha := (h, <1“ , <![, A), where := + l,u>)} and <5 := "d-

Hb := {h, < 2» where < } := < “ and <2 := < -1

Hc := (h, <3 j , < 2j A), where < f := <J f U <3—1 and <2 := <•

We need some more notations. For an n-frame Q =■ (g, (<l,|t < n),A ) define the i-projection by

P ri(G ) := (5, < i ,A ) .

For an n-modal logic A let F r,(A ) denote the fragment of A in the language with

Lemma 5.0.2 Ford € {a, 6, c } Hd is a descriptive Z-transitive frame and(1) Fr{Sf{Ha)) = Fr((fö2G.Z).Id),(2) F r(S f(H h)) = F r ( { ( « , < , >)|n € « } )(3) F r(S f(H c)) C Fr(K .B a ® K ).

Proo f. That the Hd are descriptive 3-transitive frames can be proved by standard methods. (1) Clearly Fr(S f(H a)) 2 Fr((<82G.Z).Id). Now suppose that g € F r(S f(H a)). Then P ri(g ) € F r(S f(P riH a)) and, by Lemma 4.3.1, P rx(g) € Fr(G .3). Suppose that

72 5 SF-COMPLETENESS IN POLYMODAL LOGIC<1 ^ < 2. Then, since < “ 3 there are x i,X2 € g with xi < i but not x\ <2 *2- By 3-transitivity of Ha and CoroDary 2.4.13 there exists a p-morphism p from a subframe of Ha onto a subframe f of g with x\,x-2 € /. Clearly p_1(x 1) = {w + 1} and w € p-1(x2). But then, since u> <3“ u>, we have p_1(x2) Q 0 1p_1(x2), which is obviously impossible. (2) and (3) can be proved analogously.

Proposition 5.0.3 (1) K.B\ is an iterated Sf -splitting of K and strictly Sf-complete but K.B\ ® K is not strictly Sf -complete.(2) Let A £ SK2-t with Pr\(A) C G. 3. Then A is not strictly Sf-complete above ÄY(3) Let A be a bimodal subframe logic with A C $ 26’ .3. Then A.U\ and A.Id are not strictly Sf-complete above ÄV

Proof. (1) Define A := Th(Sf(Hc)) n K.B\ CO K. Clearly A C K.B\ C*0 K and, by the Lemma above, Fr(A ) = Fr(K.B\ C>0 K). (2) If A is incomplete there is nothing to show. Otherwise define 0 := Th(Sf(Hb)). Clearly 0 2 N^.t and therefore 0 n A C A. But F r(A ) 3 Fr{G.Z C?) G.Z.t) 3 F r (0 ) and therefore F r(A ) = Fr(A fl 0 ). (3) can be proved analogously. H

This proposition shows that, contrary to the transfer results of K racht & W olter [91] concerning completeness, the map A •-» A 0 K does not preserve ^/-completeness. (3) shows that there is a large class of bimodal subframe logics A with Qip <-* Ü2P £ A but whose Kripke-frames satisfy <1 = < 2- Remember that above &)2T there is no such logic. We will now restrict our investigation to connected logics.

5.1 C onnected Log ics

For a map jrn : {1 , . . . , » } — ► {1 , . . . , n } with jr„ o 7rn = Id define CV„ := {p — □«'0Tn(,)p|t < n}.

It is easy to check that Kn. Cirn is an Ä-persistent subframe logic with F r(K n.C*n) = {h€ FrlVi : << = < - * } .

If the dimension is clear we will omit the subscript n ana write 7r instead of jt„.

Examples.(I ) (#nK.Bi = Kn.Cidn, where Idn denotes the identity on { l , . . . , n } .

(II) Define p : {1 ,2 } — * {1 ,2 } by p (l ) := 2 and p(2) := 1. Then K2.t = K2.CP.

A simple filtration argument shows

Proposition 5.1.1 For any jt the logics /fn.Cw,® n T.C*,®nK4.Cv and ®nS4.C* have the fmp.

Definition 5.1.2 An n-modal logic A is connected if there exists a ir with Kn.Cv C A.

5.2 Sf-splittings in Lattices of Connected Logics 73If we call an extension of K 2I a tense logic then all tense logic are connected logic. Other examples of connected logics are the logics ($nS5. By Proposition 5.0.3, K^.t is not strictly ^/-complete above K<i and it follows that in general the logics Kn.Cv are not strictly 5/-complete. But in presence of <X)nT we have

Proposition 5.1.3 For all tt with ir on = id the logics ($hT.Ct arc iterated Sf-splitting of Kn and strictly Sf-complete.

Proo f. Since m ’T is an iterated Sf-splitting of Kn the proposition follows if (#nT.Cw is an iterated 5/-splitting of C*OnT. Define for i < n:

T ':= ( {0 , l } , ( < l j| j< n » ,

£' := ({0 ,1 }, (r*|ji < n)),

where <*{(0 ,0 ),(1 ,1 )} : i ? j {(0 ,0 ),(1 ,1 ),(0 ,1 )} : i = j

, and r'j f {0 ,1 } x {0 ,1 } : j £ » (* )\ {(0 ,0 ), (1 ,1 ),(0 ,1 )} : j = jt( i )

Clearly (Q ')- C (T ’ )fle/- By Theorem 3.2.20, all sets (£ *)-, t < n, define an iterated ^/-splitting of &nT and C*5nT/5 (C7‘ ) - is Ä-persistent. It foDows immediately that

®nT.C* = ®nT/sfF, where F := U {(£ ’ )-|i < » }• H

It follows that (® 2T ).f and ®nT.B i = C*)nB, where B denotes the Brouwer-axiom, are strictly Sf-complete.

5.2 .S’/-splittings in Lattices of Connected Logics

We now prepare some rather general splitting-theorems for lattices of connected logics. Fix a n n € w and a map v : { l , . . . , n } — ► { l , . . . , n } with v o jr = Id. For an n-tree T = (t, (< t- |t < n)) with root 0 and x £ t define x - := {y £ t|x £ T r r (y ) } .Now dehne T (x ) = (t, (< f |i < n)) by

V < X z ;f / y <i 2 aod {^ y } % * - or' \ y < j z and { z, y } C x^ and it ( j ) = i.

T (x ) is an n-tree with root x. It is easy to show that

(I ) For all £ € F r(K n.Cw) : T < „ G iff.T (x ) < * Q.

74 5 SF-COMPLETENESS IN POLYMODAL LOGICTheorem 5.2.1 Let A be a connected R-persistent subframe logic and Q £ (T ) such that F r( A ) C {h e Fr\K g h for K, £ £ < }. T/ien £ Sf-splits A and A/5/£ is R-persistent.

Proo f. Let (7 = {g, (r,|i < n)) and let 0 be a root of Q. Define Fg := {6}. We check conditions (R I) and (R II). (R I) follows from Corollary 3.2.14. For (R II) suppose that H £ Rfr{A ) and Q ~ hb for a b C h. Let {6y|y € <} recognize (T <o G ^ h ) in H. By induction on dp^-(x) we define a set {a ^ y £ <} C A. Suppose that ay is defined for dp“ (y ) < dpj{x). If x = 0 define ao := b0. Otherwise let (t ,y ) be the pair such that y x T x< and an a C h with y € a such that there is an isomorphism p\ : F\ — ► ha with p\{z) £ a. for z £ x -. But then it is clear that there exists an F > x T (x ) and a &i C h with y € b\ such that there exists an isomorphism p : F — ► /if,, with p{y) € by for y £ t. Hence, by (I), ( T <o F •zl hh ). Now {&y|y € <} recognizes {T <o G — hb) and therefore F <o G. Hence, because H £ Rfr{A), G — F and p is a isomorphism from G to A*,. It follows immediately that p(y) £ ay for y £ t. Now suppose that xr,2. Then y = p(x ) < t- p(z) £ az. Hence y £ Ota2.H

Note that most of the G £ (T ) are not in F r(K n.Cv) and therefore Sf-split Kn.Cw but without effect. We will often have the situation that, for notational reasons, it is more convenient to work with sets containing superfluous frames than to omit these frames.

C orollary 5.2.2 Let T be an n-tree and F C (7 } be T-closed. Then F defines an iterated Sf-splitting of Kn.Cv and Kn.Cvj s F is R-persistent.

Examples. For a monomodal frame F — (/, r) define

:= {</» (r«l* £ n» lr = U-W* < n}}

and for a set of monomodal frames F define F n := \J{Gn\G € F }. Now consider a set F £ {Im , T rJ ,}. It follows immediately from Corollary 5.2.2 that F n defines an iterated ^/-splitting of Kn.Cr and that A’n.CT/s^Fn is fi-persistent. We have

Kn.C J sf{T r^ r = K n -C ^ u W p ^ U 'E 'p fS )

an this is the smallest m-transitive subframe logic above K n.Cv. The strict ^/-completeness of these logics above Kn.Cv contasts to the result proved in Chapter 4 that K .Trn,n > 1,

5 .2 S f‘Splittings in Lattices o f Connected Logics 75

is not strictly ^/-complete. It is worthwile to note that these logics are proper extensions of (®nK.Trm).C *.

)n is the smallest n-modal logic A above Kn.Cir such that in all A-frames (/, ( < t|2 < n)) all x € / have less than m -f 1 incomparable successors with respect to the

relation U { »12 < w}*

For a monomodal frame Q := (g,r ) define

Extn(G ) := {($ , (r,*|i < n))\r = n , rt C g x g for z' / 1}.

For a set of monomodal frames F define Extn( F ) := U {£ x*n(£)|£ € F }. For a monomodal formula let denote the corresponding formula in the language with

Corollary 5 .2 .3 Let T be a tree and F C (T ) be T-closed. Then Extn(F ) defines an iterated Sf -splitting of Kn.C\ and Kn.Cv/ sf Extn(F ) is R-persistent. If F defines an iterated Sf-splitting of K then

(K /V F ® K n-, ).C * = Kn.C*/s'E x tn(F ).

Proof. It follows immediately from Corollary 5.2.2 that Fxtn(F ) defines an iterated Sf- splitting of Kn. C and that Kn.Cr/ sf Extn(fF) is Ä-persistent. Now suppose that F defines an iterated 5/-splitting of K. Then F r (K /sfF $ = F r (K nC-n/s Extn(F )) andKn.Cv/Sf Extn(F ) is strictly 5/-complete and Kn.CT/ Sf Extn( F ) D (K /S*F C$ Kn-i).Cn. It follows that the logics are equalH

Examples. For (<£, F ) € {(/TO, I " ), (5 m, B “ ), (T rm, TV“ ) } the set Extn(F ) defines an iterated 5/-splitting of Kn.C^ and Kn.CT(Pr\(<f>)) = Kn.CTf s Extn(F ). This contrasts to the result that only defines an iterated 5/-splitting of K.

Remember the definition of 2m

Extn( ( I m) ) defines an iterated 5/-splitting of Kn.CT and Kn.Cv/ Sf Extn( ( I m) ) is R- persistent. This is the smallest logic A above Kn.CT such that for all A-frames h all x € h have less than m+ 1 <31-successors different from x. We will use this logic to show that ( K.Altm & Kn-i ).C \ is an iterated Sf- splitting of Kn.CT.

Given a finite rooted monomodal frame Q which 5/-splits a monomodal logic A the question arises whether Extn(Q ) defines an iterated ^/-splitting of (A & K n -^ .C * and, if this is the case, whether K /S*G ® Kn^ .C T = Kn.Cv/ SIExtn(Q ). This does not hold in general as will follow from the results of Chapter 6.

76 5 SF-COMPLETENESS IN POLYMODAL LOGICTheorem 5.2.4 Let A be an R-persistent monomodal subframe logic and suppose that T <o G € Fr(A). Then both A/S*G and Kn.C„/s Extn(Q) are defined and

(A/S*G C* Kn-ifiC* = A c* Kn-i.C*/* Extn(G) if

(R I)+ (R I) holds with Fg = {G} above A.

(R II )+ For all H £ Rfr(A): If G — hb, then there exists a set {6y|j/ £ g} C A which recognizes (G — hb) such that

(+ ) for all F >o T , 6i C h and isomorphisms q : F — *• hbx with q(y) £ by for y £ t we have F — G-

Proof. Let T = (t ,< i) and define T (n ) := (*,(<,• | < ,= {0 } for i / 1)). Clearly Extn(G) C <T(n)>.

Claim. Suppose that F £ Extn(G) and F := {K. £ Extn{G)|AJ <o F ) defines an iterated .S/-splitting of (A C*0 A'n_i ).6V and Ai := (A $ A'n_i ).C\/S F is A-persistent. Then F Sf- splits Ai and A\/S F is A-persistent.

Define F? := {F }. We check the conditions (A/), (A//) for (F , F?) above Aj. ( R I) follows from (R I)+ for G above A. ( R I I ) Suppose that N £ Rfr(Ai) and F n hb for a b C h. Then we have Q ~ Pr\{hb) and Pr\(H) £ Rfr(A). Now let {6y|y £ t } satisfy the conditions of (A77)+ for (G — Pr$hb)) in Pr\(H). By Lemma 3.2,5, there is a set {c y|p £ t} C A with Cy C by such that {cj,|y <E t } recognizes (T (n ) <o F ~ hb) in H. Clearly {cy\y £ t } still has property (+ ). By induction on dp -(y) we define a set {a y|p € t} C A. For y = 0 let ay := Cy. For y ^ 0 take the z £ t such that z <\ y and define Oy := Cy Cl O*(i)a2. We prove that {a^p € t } recognizes (F c; hb) in H. Again the interesting step is (AO ). Suppose that x £ t and y £ ax. An argument similar to the proof of Theorem 5.2.1 shows that there is a K >o T (n ) and hj C h with y £ b such that there is an isomorphism p : K — ► h^ with p(z) € c2 for z £ t. It follows from (+ ) that P t\(K,) — G, and then, by definition of Ai and because {cy|p £ f } recognizes (T (n ) <o F a hb), that K. = F. Clearly p(z) £ az for z £ t. Now suppose that xr,z. Then y — p(x) < t- p(z) £ az and therefore y £ O,a*.

It follows by induction that Extn(G) defines an iterated 5/-splitting of (A ® Kn-$.CX such that (A ® Kn-i).C„/s Extn(G) is A-persistent. The claimed equality of the logics follows immediately.H

Examples. (I) Define Fm = (/, r) — and Km := (/,/x /).

Clearly fCm € (Tm). Define A := A(ü^1 V;c —► ->po). Then, by the example above, (A® Kn-i).Cw = Kn.Cr/VExtndln)). It is easy to check that the set

:= {G\Fm—1 <o G So A)to_j) .defines an iterated 5/-splitting of A such that (A/)+ and (A//)+ are satisfied for all steps. It foDows that Extn(F) defines an iterated 5/-splitting of (A & A'n_1).C, and

5.2 Sf-splittings in Lattices of Connected Logics 77

that (A / S' F 0 Kn. ! ) . C , = (A 0 Ku ).Cr/Sf Extn(F). Now A /s/F = K.Altm. Hence

( K.Altm & Kn-i).Cr is an iterated 5/-splitting of Kn.C*■.

(I I ) Define G2 := | x-—x| and G3 •= 1 •—»x | . W e have £ 2 , £ 3 € ( 1 x—»x [ ). It is easy to check

that Qi satisfies the conditions (A / )+ and ( RII)+ above K.Tt\ and that £ 3 satisfies these

conditions above K.Tri/^ Q2. It should be clear that A'4 = K .Tri/sfG2/sfG3. Hence

(A4 0 A'„_i ).C„ = (K.Tt,I sIQ2Is1Q3 0 A'n_i ).C,= (A '.T r , 0 A n_ ,).C V / s/Extn{Q2)lsiExtn(63)= Kn.C*!s} £ x tn( T r r ) / ^ Extn(g2)/ Sf Extn(Q3)

It follows that K4.t is an iterated ^/-splitting of K2.t and strictly ^/-complete above

K2.t. Note that this result contrasts to the result that A'4 is not an iterated 5/-splitting

of K by finite frames. Another example is c>on55. oonT.B\ is an iterated ^/-splitting of Kn and (X)n55 = oon A’4.T.f?i is an iterated 5/-splitting of (X)w7\f?i. Hence cxOn55 is an

iterated 5/-splitting of Kn and strictly Sf-complete. W e will later investigate the lattices

of extensions of these logics in more detail.

C o ro U a ry 5.2.5 The logic K4.t is an iterated Sf-splitting of K2.t by finite frames.

A note on the Universal Modality

Goranko & Passy [92] study the extensions o f (K $ Sh).U2• W e can axiomatize this

system by (K & 5 4 ).C P with p : {1 ,2 } — ► {1 ,2 } with p (2 ) = 2 and p ( l ) = 2. Hence for

* = 1 , 2 we have p —► 0{02p € (K &SS).U2 but we do not have pop = td as was required in

the definition of connected logics. A t first sight this should not make any difference for the

study o f strict 5 /-completeness, but it turns out that it o it = id is exactly the reason for

so many strictly 5/-c.omplete logics: It follows from the results in Goran ko & Passy [92] that a monomodal subframe logic A is strictly 5/-complete above K only if (A $ 5 4 ).C P is

strictly Sf-complete above ( K <&S4).CV. It follows that for instance (K.Trnöo S4).CP and

(A '4 ® 5 4 ).Cp are not strictly 5/-complete above (K ® S4).CV. But in case o f a 7r with

7r o ic = id the logics (Trn ® 54).CV and ( A'4 & 54).CV are strictly 5/-complete even above

K2.C*. These examples show that the proofs o f Theorems 5.2.1 and 5.2.4 make essential use o f the fact that the inverse modalities are uniquely determined in connected logics. Nevertheless (K #)S5).U2 is 2-transitive and therefore any finite rooted 2 -fram e splits and

5/-splits (A ' &> 5 5 ).U2.

78 5 SF-COMPLETENESS IN POLYMODAL LOGIC5.3 The upper part of SA fn

Theorem 5.3.1 For all n,l £ w all subframe logics above <$nK.Alti have the fmp.

Proo f. Let A be a subframe logic above (tinK.Alti and ~'<f> £ A. There are a A-frameg , x € g and a valuation ß with g,ß,x f= <f>. Define m := dg{4>) and b := T r ”*(x). Clearly <7i is finite. Define a valuation 7 by 7(p) := ß(p) D b for p € Var. It is easy to check that

h , 7,x t=

Theorem 5.3.2 For all n,l £ all subframe logics above 0QnK.Alti are strictly Sf- complete and have strict Sf-fmp.

Proof. We first prove strict S/-completeness. For a frame Ti = (A, (<;|i < n), A) let « : = [ ) { < ,|z < n }. The proof is based on the following simple definition.

Defin ition 5.3.3 Let Ti = {h, (<f,|i < n),A ) £ Rfr. Then Ti is weakly smaller than m, in symbols ||?f|| < m, if |Tr£(x)| < m for all points x € h. 7/117411 < m for a b £ A we write ||6|| < m . We write ||&|| = m if ||6|| < m and ||&|| ^ m — 1. A class of frames F is weakly smaller than m if ailTi € F are weakly smaller than m, we write ||F|| < m.

Strict Sf-completeness is shown if for all » , m £ u the sets

:= {Ti £ Rfr{Kn)\\\H\\ £ l for all / € u; and ||Fr(S/(W))|| < m }.

are empty. Note that for Ti £ H™ and Tt\ £ Rfr(Sf(Ti)) with ||74|| £ m we haveHi € H™. Define $ n := {p —► üj V (^>iPÜ ^ »)|* ^ n }. It is easy to check that

{Ti € Rfr(Kn)\<i C for t < n } = Rfr(Kn( * n)).

Claim 1. For all n, m € w and Ti £ HJJ* we have Ti \£ # n.

Proof. Let Ti £ H “ . There exist z , x i , . . . z m € h with x < x; for t < m. Chooseb = {x , x j , . . . , xm} such that £ (| < « n62||t < n) is minimal. Choose a cycle free K, <x hb and {cy\y £ 6} C A which recognizes (/C <x hb) in Ti. Now define

aXi := cx- n f|{^>i c*la;« * ,/ < n } for t < mand ax := cx. It follows with the methods of the previous chapters that {a y|t/ £ 6} recognizes hb in Ti tf Ti \= 4>n. But hb £ F r(S f(T i)) and therefore Ti ^ $ n.

We need some notations for generated subframes. For a frame Ti = (h, (<j,|t < n), A) and x £ h define Tix := (hx, (<,-nfc2|t < n), Ax), where hx := T rh{x) and Ax := {a flA ^a € A).

Claim 2. Let Ti £ H™ and 61, 62 € A with ||6j|| < m for t = 1,2. Then ||6j U 62|| < m.

Proof. Let 6 := 61 U 62 and suppose that ||6|| % m. Then we have ||?4 || ^ lb for all k £ w. We may asumme that 6j Cl &2 = 0 and that there are x £ bt and xa, . . , ,x TO £ b2 with x < xj for j < m. Now choose a = {x ,x 1?.. . ,x m} with these properties so that

5.3 The upper part of SAfn 79<,• na2||t < n) is minimal.

Let AJ be cycle free with K <x ha. Choose {c y|y £ a} which recognizes (K. <x ha) in H with cx C and cy C b2 for y £ a - {x } . Let cx := |J{cvly € a - { x } } . Choose cC b such that there exist p : hc ~ ha with p_1 (y ) € cy for y € a such that

53(|Tr\(z) n ci||z € c - p-1(x )) >s maximal.

Define z := p_1(x ) and for y £ c - { z} define c(y) := Tr/t(p) n c\. Choose cycle free frames K.y <y hc(y) for y € c - {z} and choose sets {c\\z £ c (y )} C A which recognize (AJy < y /ic(v)) in H with c\ C cy for y € c — { 2} and c% C c\ for z £ c(y). Let

a. := cx n n {0 «c$ l* < i P ( y ) , y € c - { z } , t < n}.

Define H1 := 7id with d := az U |J{cy|i/ € c — {z}}. Consider the frame Hi. It is tedious but easy to show that { az fl Tr/ld(z)} U {c y D Trhd{z)\y £ c — { z} } recognizes hc in Hi. But then hc £ F r(S f(H )) and ||/ic|| ^ m and this contradicts to H £ H™.

Claim 3. For all H £ H™ and x £ h such that x has more than 2m successors there exists a set 6 C Tr\{x) with |6| > m such that Hy € H™ for all points y £ b.

Proof. Suppose that there is an H £ H " ‘ and a point x £ h such that |Tr^(x)| > 2m and |c| < m for c := {y £ Tr\(x) — {x }|Hy £ H™}. Take a set b C Tr\(x) — ( { x } U c) with |6| = m such that Hy H™ for y £ b. Then choose a set cj D c with Cj £ A and x £ ct, ct fl 6 = 0. Let H 1 := Hh-a - Consider H\. = (g, B). Then ||<7y|| < m for any point y £ g with y x and 7Y*(x) > m. We have Hx £ H™ and obviously { x } £ B. Now ||{x}||, ||g — {x}|| < m and therefore, by Claim 2, ||Wi|| < m. We have a contradiction.

Now suppose that there is an n £ u with U{HJT|m € w } ^ 0. Let N be minimal with this property. We show that ^ 0 leads to a contradiction.

Claim 4. Let H £ Hjy and x £ h with x-jJ,x for an i < N . Then there is a set a € A with x £ a and ||o|| < m.

Proof. Let x £ h and x ^ x . Choose a £ A with x £ a and a fl 0 ;a = 0. Suppose that ||a|| % m. Then Ha £ H # and Ha |= □ t±. Let Ha = (ha,(< j\ i < n ),B ). It follows immediately that (ha, (<l j|j < n ,j i},B ) which contradicts the assumptionthat N is minimal.

C laim 5. There is an m € n; and H £ with H £ Rfr(<&NT).

Proof. Suppose that the claim is false. Then choose H £ with

M := max{||6|| € w|6 € A ,b^ 0 } minimal.

80 5 SF-COMPLETENESS IN POLYMODAL LOGICIt follows from Claim 3 that there is an x € h such that there is a set b C Tr\(x) — {x} with |6| = m and Hy € H]y for all y € b. It follows that Hy ^ C%NT for y € 6. Hence, by Claim 4, there exist sets ay € A with ay fl hy ^ 0 for y € b and ||ay|| < m. Take with 7V*(y) H ay ^ 0 for y € b. Now define for y 6 b by induction:

do(y) := LK I* < N } - ay.

dj+M := U { o . d i ( y ) K < ^ } - ^ ( y ) -

We have 6 U {x } C d := U {°v ly € 6} U U {^ j(y )ly € 6, j < k + 1}. We check that lldi(y)ll < m for ail j < k + 1 and y € b. Suppose that this is shown for ail j < n and suppose that ||dn(y)|| £ m'i we assume that n > 0; the step « = 0 is proved analogously. Then it follows from the definition of M that there is a c C dn(y) with c € A and ||c|| = M. Take a z\ € c with ITrJ^zi) Cl c| = M. Now there is an i < N and z2 € d „_ i(y ) with z\ < iz 2. But then ||dn_i(y)Uc|| < m by Claim 2 and ||dn_i(y)Uc|| ^ M. This contradicts to the definition of M.It follows again from Claim 2 that ||d|| < m. But this is false, since 6 U {x } C d. We have a contradiction.

Define KJ& := H # n Rfr(®NT).

Claim 6. There are m £ u and H € with(+ ) for all points x for which there is a y with x <J y and y<fi x there is a set a € A with x € a and ||a|| < m.

Proof. Suppose that there is no such H. Let H € K $ . Take x ,y and to < N with x <y0 y but y- j x such that x is a counterexample for (+ ) . Then take sets ao,&o € A,xq € oo, with oo —► O,0 6o = h and 6o H Ojao = 0 for 3 < N. We have ||ao|| jf m. Hence we can choose tj < N and x j ,y j € ao with xi <1,-, y i but yi^Jxi such that x\ is a counterexample for (+ ) in Hao. Choose ai,&i C ao in A with x i € oi and ai —*■ 0;,&i = h and 6j n OjOj = 0 for j < N. Again we have ||ai|| ^ m. We go on in this way and get a sequence 6* € A, k < m + 1 with 6m+i -*■ Ojkbk = h and bkOOjbi = 0 for k < /, j < N and 6,n 6* = 0 for / ^ k. Take xjt € 6* for k < m + 1 such that xTO+i < IJt x* for k < m + 1. Let 6 := { x 0, .. . ,x m+1}. Clearly we have ||h6|| ^ m and hb is T-cyde free. Hence hb 5/-splits C$NT and <$NT/Sfhb is Ä-persistent. But then we have hb € Fr(S f(H )). This contradicts to H e K $ .

Now let H € KJv with property (+ ) such that M := max{||b|| € w|6 € A,b ^ 0 } is minimal. It follows from Claim 3 that there is an x € h such that there is a set 6 C Tr\(x) — {x } with |6| = m and 7iy € for all y € b. Hence, by Claim 1 and property (+ ) of H, there exist sets ay € A with ay fl hy 0 for y € b and ||ay|| < m. Take a k € u> with 7V*(y) n ay £ 0 for y € b. Define dj(y) as in the proof o f Claim 5 and let d := U{°y|y € 6} U U K (y )| y € 6 ,i < fc + 1}. It follows as in the proof of Claim 5 that ||(f|| ^ M and ||(f|| < m. Again we have a contradiction to the definition of M .

We prove strict S/-fmp. Let A D $)nK.Altm and let A i 6 SMn with F r j{A ) = F r j (A j).

5.3 The upper part of SAfn 81Suppose that Ai C A. Then A] 2 &>nK.Altk for all A; € u. It follows from H * = 0 for k € w that there is a frame h € F r (A j) with h ^ <$nK.Altm. Hence there is a finite subframe g of h with g ^ K .Altm.-\

Corollary 5.3.4 All tabular subframe logics arc strictly Sf -complete and have strict Sf- fmp.

Definition 5.3.5 Let C be a describable operation and let A be a C-logic. Then A is C-pretabular if A is not tabular but all C-logics 0 D A are tabular.

The following proposition follows immediately from Zorn’s Lemma and the fact that tabular logics are finitely axiomatizable.

Proposition 5.3.6 All non-tabular C-logics are contained in a C-pretabular logic.

With the help of the following easily proved lemma we get all desirable properties of the upper part of SAfn.

Lem m a 5.3.7 Suppose that A is a subframe logic such that there is a rooted frame g € F r (A ) with |</| > m. Then A has co-dimension > m in SAfn.

Corollary 5.3.8 All Sf-pretabular logics have infinite co-dimension in SAfn.

Corollary 5.3.9 An n-modal subframe logic is tabular if and only if it has finite codimension in SAfn. For an n-modal tabular subframe logic A it is decidable whether

Corollary 5.3.10 All Sf-pretabular logic have the fmp.

It is shown in Blok [80a] that there are 2H° pretabular logics above KA. This contrasts to

Theorem 5.3.11 A monomodal subframe logic A not above K.Altn for all n £ u is an Sf-pretabular logic iff A is one of the following logics: A € {G .3 ,G rz.3 ,S5 } or

Kn(<t>Sf) = A.

n±L

1|n € u>} or

8 2 5 S F -C O M P L E T E N E S S IN P O L Y M O D A L LO G IC

A = Th(Sf(F2)) with F2 := {


|n G u } or

| n € u;} or

A = Th(F^) with F4 := { |n G a;} or

A = Th(Sf(Fs)) wtft Fs := {



A = Th(Sf(Ft)) with Fs := {

R7

R7

R7I

R7

n + l

n+1

n + l

In G u>} or

|n G a;} or

|n G u>} or

|n G u>}.

Proo f. It is easy to prove that all the defined logics are 5/-pretabular. Now suppose that A is 5/-pretabular and A 2 K.Altn for a l lngui. We know that A has the fmp. The proof is based on the following easily checked observation

Observation. Let ((<?„, < „ ) ) new be a sequence of finite frames with |<7„| > n for all n G w. Then there is a sequence ((/im, rm) )mew with |/im| > m for all m G w such that all (hm, rm) are subframes of a frame (gn, < „ ) for an n G u> and

(1) rm = 0 for all m G u> or

(2) rm = {(x ,x)|x G Ato} for all m G u? or

(3) <hm,rm) ~ ( {0 , . . . ,m },s m) with sm D< for all m G w.

A has fmp and A 2 K.Altn for all n € w. Hence there is a sequence ((<?»»»<*»»*»))new with A = Th(Sf({gn\n G u>})), |$n| > n and gn = T r ]n(x „ ) for all n G w.Case 1. There is a sequence (AnOmgo, of subframes of (gn — {* « }> < n )»n € w, with property(1 ) . It follows immediately that A is one of the logics Th(F\),Th(Sf(F2)),Th (S f(F7)) or Th(Sf(Fs)).Case 2. There is a sequence (hm)m€w of subframes of (gn — {x n} , <Jn),n G oj, with property(2 ) . It follows immediately that A is one of the logics Th(Sf(F3)), Th(F4), Th(Sf(Fs)) or

5.3 The upper part of SAfn 83Th(Sf(Fe)).Case 3. There is a sequence (Am)mgw of subframes of (gn — { x n}, <]„}, n G w, with property (3). In this case A = 7 7 i(5/ ({({0 ,... ,m },s m)|m G w } ) ) with sm D< for all m G w. Consider the sequence ( ( { 0 , . . m},sm— <))mgu/-Case A. There is a sequence ( fm)m of subframes of ( { 0 , . . m},sm— < ) ,m G u>, with property (1). Then we have A = (7.3.Case B. There is a sequence (/m)tnew of subframes of ( {0 , . . . , m } , s,n- < ) ,m € w, with property (2). Then we have A = Grz.3.Case C. There is a sequence (/m)T O of subframes of { {0 , . . . ,m}, sm- < ),m € w, with property (3). Then A = 77i(5/({ ( {0, . . m}, € w } ) ) with <Jm D for all7/1 G a;. Hence A C 55. But then A = 55.H

It follows that there are exactly 7 5/-pretabular logics above A'4 and that an 5/-pretabular logic A is above K.Trz if and only if it is above K.Trn for an » > 1 if and only if it is not above K.Altn for all ti G o». Let us note that is easy to show that there are at least Ko 5/- pretabular logics above K.Alt-2. We conjecture that there are precisely Ko 5/-pretabular logics in Af but lack a strict proof.

Corollary 5.3.12 There exist Sf-pretabular logics which are not pretabular.

Proo f. Th(Sf{F2)) is not pretabular. H

The following theorem shows once more the fundamental role of the logic G\3 for investigations on strict 5/-completeness.

Theorem 5.3.13 All monomodal Sf-pretabular logics not equal to G.3 are strictly Sf- complete.

Proo f. We know that all subframe logics above K.Altn for an n 6 w are strictly 5/- complete. Further it follows from Theorem 4.4.1 that <7.3 is the only non-strictly 5/- complete logic above A'4. Hence it remains to show that the logics T/t(5/(F,)) with i = 5 , . . . ,8 are strictly 5/-complete. Let A be one of these logics. Clearly we have A D K.B\ = A'/^Bj". Define 7-j := ( { 0 , 1,2,3 },<). Now K.B\ is connected and therefore (7-j) defines an iterated 5/-splitting of K.B\ such that 0 := K.B\/S (T$) is Ä-persistent. We have A 3 0. It should be clear that V (0 ) is a locally finite variety and it follows that all extensions of 0 have the finp. Now 0 is an iterated 5/-splitting of K by finite frames; hence^jill subframe logics above © are iterated 5/-splittings of K by finite frames. It follows that all subframe lögics above 0 are strictly 5/-complete.H

8 4 5 S F -C O M P L E T E N E SS IN P O L Y M O D A L LOG IC5.4 Tense Logics

A bimodal logic A is a tense logic if A D K2-t. We denote the modality ü^C^) by □ + (D_ ). For a monomodal formula 4> the translation of <p into the language with □ + (m- ) is denoted by For a monomodal logic A define A + .t := (A $5 K).t and A~.t := (A' M A ).t.A +.t is called the minimal tense extension of A. A bimodal frame Ti = (h, < i, < 2, A) is a tense frame if <31 = < J1 • Hence in a tense frame the relation <d 1, which will be denoted by <3+ , uniquely determines the relation <2 ( denoted by < “ ) and vice versa. In a tense frame the operation üi (CI2) is denoted by D+ (□ “ ). Let Ti = (A, <3, A) be a monomodal frame such that {x € A|Vj/ : y <1 x y £ a } £ A for all a € A. Then the tense extension Ti1 of Ti is defined by

TP := (h, where <J+ := < and < := <

It is worthwile to note that one has to take some care by switching between tense frames and monomodal frames: Clearly for every Kripke frame (h, <1) the structure (h, <J, <J—1) is a tense frame, but for a general frame (h, <], A) the structure (A, <d, <]-1, A) need not even be a frame. The existence of such a monomodal frame follows immediately from the next theorem and the fact that there are non strictly 5/-complete logics above K4. The 1-projection (2-projection) of a tense frame Ti is denoted by Ti+ (Ti~). For a class of tense frames K let K + denote the class of 1-projections of the frames in K . For a tense logic A let Fr+(A) denote (F r(A ))+. The 1-projection of A is denoted by A+. Note that F r+ (A ) C FY(A+ ) but that equality does not necessarily hold. A p-morphism between tense frames is called a t-p-morhism.

It is easy to check that the map ( ).t : SK — *• SK.t, A *-► A+ .t, is a V'homomorphism. Kracht [92] assumes that this map is injective or equivalently, that (A +.t)+ = A for all A € SK. It follows immediately from the following theorem that this is not the case.

Theorem 5.4.1 For all subframe logic A above K 4

{0 € SK\(e+.t)+ = A } = Fnsfc(A).

Especially we have that {0 € SK\(0+.t)+ = A } = {A } iff A is strictly Sf-complete.

Proof. First note that F r ((0 + .t)+ ) = F r (0 ) for any monomodal logic 0. Hence (0 + .<)+ = 0 if 0 is complete, especially if 0 is a subframe logic above KA. The inclusion ” C" follows immediately from this observation. Now suppose that (0 + .t)+ / A but F r (A ) = F r (0 ). We first show that 0 + .t D A'4.t: Suppose that this is not the case. Then there is a refined non-transitive tense frame Ti with Ti [= 0+.f. Now any g € F r (0 + .t) is transitive because F r (A ) = F r (0 ). Hence 0 + .t Cl K4.t is a subframe logic not equal to KA.t with the same frames as KA.t. But this is impossible since KA.t is strictly 5/-compete above K.t.

5.4 Tense Logics 85

Hence (0 + .t)+ is a subframe logic above K4 with F r ( (0 + .t)+) = F r (A). But then both logics are equal since they are complete. H

It follows from Theorem 4.4.3 that for all subframe logics A with A'4 C A C G the cardinality of {0 € 5A'|(0+ .t)+ = A } is 2H°. Let us state the main theorem of this chapter.

Theorem 5.4.2 Let A be a monomodal subframe logic above K 4. Then the following conditions are equivalent:

(1) A is strictly Sf -complete above K2.t.

(2) A+.t is an iterated Sf-splitting of K2.t by finite frames.

(S) A+ .t is an iterated Sf-splitting of K4.t by finite frames.

(J[) A is strictly Sf -complete above K4.t.

(5) A is elementary.

(6) A+.t is elementary.

Before starting the proof let us note some consequences. One conclusion of our investigation on strict 5/-coinpleteness in monomodal logic was that there is no direct connection between this concept and other concepts of modal logic. The situation in tense logic is completely different. The theorem states that at least for a quite natural class of tense logics strict ^/-completeness is equivalent to some well-known concepts of modal logic, namely to elementarity and therefore, by Theorem 3.1.5, to compactness. The following theorem will follow immediately from the proof of Theorem 5.4.2 and Proposition 5.1.3.

Theorem 5.4.3 Let A be a monomodal subframe logic above 54. Then the conditions (1), ..., (6) are equivalent with K2.t replaced by K2 and K4.t replaced by S4.t.

Note that in presence of Proposition 5.0.3 we cannot replace K2.t by K2 in Theorem 5.4.2 because a logic A € SK2.t with A+ C G.3 is not strictly 5/-complete above K2.

Corollary 5.4.4 For any two elementary subframe logics AiyA2 above K4 the tense logic A+.AJ.t is an iterated Sf-splitting of K.t by finite frames. If A\ D 54 then A|\A2 -t is an iterated Sf -splitting of K2 by finite frames.

It follows from the results concerning connected logics that any reasonable elementary subframe logic above K4.t is an iterated 5/-splitting of K2.t by finite frames.

For the proof of Theorem 5.4.2 we need some concepts well-known from the study of logics above K4. Suppose that Q = (g, < ) is a transitive frame. A set U C g is called a cluster

8 6 5 S F -C O M P L E T E N E S S IN P O L Y M O D A L L O G IC

i f U = { x } and x- J x or V is maximal with the property Vx, y € U : x <J y. A cluster U is called degenerated if U = { x } and xjt x; U is called simple if U = { x } and x < x. A cluster U is proper if |f/| > 1. We can extend the relation <3 to clusters in the following way: For x € g and clusters U\, Ui define (1) x < U\ if there is a y € U\ with * <1 (2)U\ < x if there is a y £ U\ with y < x and (3) U\ < U2 if there are x € U\ and y € U2 with x <] y. Let Cl{Q) denote the set of clusters of Q. Then it is easy to check that Clu(G) := (Cl(G),<l) is a T-cycle free frame. If G is finite we define the depth of x € g by dpg(x) := dpCiu(g){U ), where U is the cluster with x € U.

Again we have to manipulate some frames. Suppose that G = (<7, r) is a finite transitive frame, that U is a cluster in G and that H = (h, < + , <]“ , A) is a transitive tense frame with h fl g = 0. Then the substitution of U by "W in G is defined as follows.G[H/U] := (g[h/U],s+ , s ~ , B ) with

g\h/U) := (g -U )U h .

s := (r—(f/xf/))U<lU{(x,y)|x € g -U ,y € h,x r f/}u{(x,y)|x € h,y € g -U ,U r y).

B := {a U c|a € A,c C g—U}.

Lem m a 5.4.5 G[H/U] is a transitive tense frame. G[H/U] is descriptive ifH is descrip- itive.

P roo f. Transitivity is clear. G[H/U] is a frame if B is closed under fl, — , 0 + and O- . Closure under O, —, is obvious. We show closure under 0 + . Closure under O- can be proved analogously. Let a € A and cC g -U . Then 0 + (a U c) € f l iff 0 + (a U c) n h € A since G is a Kripke-frame. But O+(a U c) n h = h if there is an x € c with U r x and O+(a U c) fl h = O*a otherwise. The proof of the second claim is straightforward. HIf this causes no confusion we will also write G[H/U] for the monomodal and transitive frame (g[h/U],s+, B).

5.4.1 Descendants and Variants

5.4 Tense Logics 87

In this chapter we define variants and descendants of finite transitive frames. Both concepts were introduced in F ine [85], but there are essential differences to our definition. First F ine’s definition of descendants is not weak enough, even for the results proved in this paper. Second we need an induced partial order <7 on the finite frames without strictly ascending chains to prove Theorem 5.4.2. This is the reason why we have to iterate the process of forming descendants and variants and thus get a closure operator on the finite transitive frames. Besides of F ine’s paper [85] the exposition is also inspired by M. Zakharyaschev’s [92a] investigation of elementary confinal subframe logics above A'4. In the following two sections we only deal with monomodal transitive frames.

Let Q = (g, r) be a finite transitive rooted frame and U € Cl(G) and let hng(U) denote the set of immediate predecessors of U in Clu(G). Suppose that \Img(U)\ > 1. Then a transitive Kripke frame T = (/, s) is an im mediate descendant of G by replacement of V if there are two clusters U\, U2 G Irng(U) and two sets V\, V2 C / such that

(1 ) / = (0 - t f ) u v , u v 2.

(2) Vj and V2 are clusters in T isomorphic to U and 1 Vj for i ß j .

(3) For all x, y € y — U : x r y x s y.

(4) U^Vu U2sV2 and Ux £ V2,U2 ß V,.

(5) For V G Cl(G), U 'ß U and i = 1,2 : VisW UrW.

(6) For U' G Cl(G), U' ß U and i = 1,2 : U'sVi =► V rU .

(7) For V G Cl(G), V ^ U: If U'rU then there is an i with U'sVi.

Note that, by (1),(2),(3), the clusters in T are (Cl{G) - {t/>) U {V j, V2}. Hence (4) . . . (7) are well-defined. We sometimes write T = G[{V\, V2}/U\), but note that T is not uniquely determined by G,U,Vi and V2.Now define operations Dj for j = 1,2,3 by T € D0(G)(D i(G ), D2(G)) iff T is an immediate descendant of G by replacement of a degenerated (simple, proper) cluster. The following figure illustrates the definition of immediate descendants. The frames to the left of a frame

are its immediate descendants.

8 8 5 SF-COMPLETENESS IN POLYMODAL LOGICA function

7 : {(G, U)\G € Frj(KA) rooted, U € Cl{G) non-degenerated } — ► u> - {0 }

is called a cluster valuation. Now let Q be a finite, rooted and transitive frame, let U G Cl(Q) be non-degenerated and 7 a cluster valuation. Remember the definition of T* = ({0 ,...n — 1} ,< ) and define T* := ( { 0 , . . . ,n - 1 },<). Then G[T*/U ],n := 7(((?, U)), is an ir-immediate variant of G by application of 7 to U.If U is proper thenG[T*/U ],n := 7((£7, U)), is an r-immediate variant of Q by application of 7 to U.We define operations KJ and V* on the class of finite and transitive frames by

F G Vj(G) iff F is an ir-immediate variant of G by application of 7.

T G V*{G) iff F is an r-immediate variant of G by application of 7.

Let D := {D0,D ^D 2} and V, := {V^V;}. For H C := {Do, Du D2,V*,V;} and n G w define

FftiG) := \J{WiW2... Wn(G)\Wi G H for i < n}

Definition 5.4.6 Let 7 be a cluster valuation, Q be a finite rooted transitive frame and H C D~,. The closure of { 6 } under H is Fh (G) := U {^w(£)ln € w}. The strict closure of G under H is Ffj(G) := U{^j&(0 )U < » G w}. The set of descendants of G is Fp(G)- The 7-closure of {G} is F~,(G) := Fq^(G) and the strict 7-closure of G is f;(G ) := Fp^(G). A set of finite rooted frames F is 7-closed if Fy( F) := \}{F {G)\G € F } C F.

It is easy to check that for any cluster valuation 7 the operation F\ is a closure operator on the set of finite rooted frames and that F*(G) is 7-closed for all G- Define relations <7 and <7 on the finite transitive frames by T <7 G iff G € F^(F) and T <7 G iff G € F *(T ). It follows that

<7 is a partial order without infinite strictly descending chains.

Any G has only finitely many immediate successors (because F*(G) is finite).

<7 is the strict partial order corresponding to <7.

Proposition 5 .4.7 For any cluster valuation 7 and any G the set (F7((7), <7) contains no infinite strictly ascending chains.

5.4 Tense Logics 89For the proof of this proposition we need some more notations. For G\ € W((/), W 6 Dy, there is a cluster U € Cl(G) such that G\ — G[S/U] for an S € { { V i , V i)-,%*■>%*}• For a sequence ({?,),<„ with Gi+1 € Wi(Gi), Wj e for i < n, we define by induction whether the occurence Wj is weakly applied to U € Cl(Go)'We say that Wo is weakly applied to U € C l (Go) if G\ = Go[S/U} for a suitable S. Wj is weakly applied to U € Cl(Go) if there is an i < j such that Gi+1 = Gi[S'/V] and Wj is weakly applied to U and Gj+i = Gj[S" /V'] for a cluster V in S'. Note that any occurence W, is weakly applied to precisely one cluster in Go- We will sometimes use the term applied instead of weakly applied.

Claim 1. Let Gb Q G be a generated subframe of G and let Gi € W$G$,G\ € W0(£ ), where W0, W\ £ IX,, Wo is applied to a cluster in 6 and Wi is applied to a cluster in g — b. Then there are a cluster valuation 7' equal to 7 on Fy(Gi) and Wq, W{ € Dy of the same type as Wo, W\ such that there is a G\ with G[ € W{(G),G-2 € Wq(G\) and W{ is applied to a cluster in g — b, Wq is applied to a cluster in b.

Proof. Easy but tedious.

Let Gb be a generated subframe of G- Then Claim 1 tells us that we can first apply operations in Dy to clusters in g — b and then apply these operations to clusters in 6. Now the following claim follows by induction.

Claim 2. Suppose that is an infinite sequence with Gi+1 € W,((7j), W, 6 D^, andlet T be a generated subframe of G- Further suppose that only finitely many occurences of W< are weakly applied to clusters not in T . Then there is a cluster valuation 7' and an infinite sequence (G{)ieu> with (7,-+1 € W-(G-), W- € Dy so that T is a generated subframe of Gq and any W/ is weakly applied to a cluster in T .

Claim 3. For any G and cluster valuation 7 the set (Fd (G), <-,) contains no infinite strictly ascending chains.

Proof. By induction on dp(G). The case dp(G) = 0 is trivial. Assume that the lemma is shown for all T with dp(F) < n and that dp(G) = n + 1. Let T := Gb with

b := {x € g\dpg(x) = 0}.Then Gg-b has depth n. (Remember that the definition of depth is not the one usually used for transitive frames. A point has depth 0 iff it has no strict successor. Hence our definition of depth does not distinguish between irreflexive and reflexive points.) Suppose that there is an infinite strictly ascending chain in Fp(Q). Then there is a sequence (£»')«eu> with G'i+i € W((G'i), W( = Dj, j = 0, 1, 2. By induction hypothesis, only finitely many occurences of W- are weakly applied to Gg-b- Hence, by the claim above, there is a chain (Gi)ieu», Gi+i € W,-((/,), Wj € D, so that T is a generated subframe of Go and any Wj is weakly applied to a cluster in T . Then the following holds for all * € w and U € Cl(Gi) •

0) If V € Cl(Gi+ i) , then \{W € Cl(Gi)\U' < Gi U}\ > \{U' € Cl(Gi+ i)\W U}\.

(Ü) UGi+1 = & [ {V i ,V2}/f/],then \{U' € Cl(Gi)\U'<GiU}\ > \{U' € Cl(Gi+ i)\U '<Gi+l Vg}\ for j = 1,2.

90 5 SF-COMPLETENESS IN POLYMODAL LOGIC(ii) follows from the definition of a descendant and (i) follows from the fact that a U G Cl(Gi) with (7*+1 = £*-[{Vi, V2}/U] has no strict successor in G%> Hence there is no U G C l(T ) such that infinitely many occurences are weakly applied to U. This contradicts to the fact that all W{ are weakly applied to a cluster in T . H

Claim 4. For any G and cluster valuation 7 the set (F^DotDlyiy(G)^<^) contains no infinite ascending chain.

Proof. By induction on dp(G)- The case dp(G) = 0 is trivial. Let dp(G) = n + 1. Define T as in the proof of Claim 3. Again by induction hypothesis and the corollary above we have: If there is an infinite strictly ascending chain in F{D0,DuVi}(9)'> t^en there is a cluster valuation 7' so that there is a chain ({?i)«€<*/> £t+i € with G { Do, D\, Vy } and Tis a generated subframe of Go and any is weakly applied to a cluster in T. If = V*, is applied to a cluster U in then D\ cannot be weakly applied to this cluster afterwards and the number of clusters to which D\ can be applied does not increase. Hence, by Claim 3, Z?i occurs only finitely many times among the From this follows that V*, occurs only finitely many times. But then, again by Claim 3, even D0 occurs only finitely many times. We have a contradiction. H

P ro o f of Proposition 5.4.7. By induction on dp(G)- The proof is anologous to the proof of Claim 4. Let us start for the induction step with a cluster valuation 7' so that there is a chain Gi+1 G Wi(£t), W{ G D y , such that T is a generated subframe of Go and anyW{ is weakly applied to a cluster in T {T defined as above.) Now it follows immediately from Claim 3 that D2 occurs only finitely many times and therefore that even Vy occurs only finitely many times. Hence there is a ; G w with Wi G {A>, ^ 1? V y } for all i > j. This is a contradiction to Claim 4.H

By an easy extension of König’s Lemma we get

CoroUary 5.4.8 For any cluster valuation 7 and finite and rooted transitive frame G the set F*y(G) is finite.

Lemma 5.4.9 Let T be a descendant of G• Then G is a p-morphic image of T .

Proof. Clearly the lemma follows if it is shown for immediate descendants. So let T = G[{V\, V2},t/]. Take maps p, : Vi — ► U,i = 1,2, onto U and define p : T — ► G by p(ar) := x for x G g - V and p{x) := p,*(x) for x G V», i = 1,2. It is readily checked that p is a surjective p-morphism.H

Lemma 5.4.10 Let U G Cl(G) be non-degenerated. Then G is a p-morphic image of both, G[(u>,<)/U] andG[(u>,<)/U).

Proof. Let U = {xo> • • • > Define p : G [ { < ) / U ] — ► G by p{x) := x for x G g - Uand p(mn + k) := x* for n G u;,0 < k < m — 1. p is a surjective p-morphism. The case

is considered analogously.H

91

Let Q = (<?,r) and 6 C g. Then 6 is downward closed in Q ifV x , t/ € g : x t y,y 6 b => x € b.

If U\, U'2 are subsets of clusters U[, U2, then we write UirU-2 iff U{rU2.

5.4 Tense Logics

Lemma 5.4.11 Let Q = (5f,r) be finite, a C g be downward closed in Q,a / 0. Let 1/ € C7(£?), U fl a = 0, with Im G(U ) = {f/i, . . . , Uk) C C7(0O). Lurt/ier let T = (/ ,*) 6e a transitive frame with f = ( g - U ) U Vi U ... U Vn, n > 1, and (I/, r) ~ (VJ, s) /or t < n and

(%) Vx,t/ € g - U : xry & xsy.

(2) Vi fVj for i / j.

(3) Vx € g - u,y € VJ, i < n : ysx Urx.

(4) Vx € g - U,y € V{,i < n : xsy => xrU.

(5) Vi < k3j < n : UisVj.

(6) Vj < n3i < k : U{ /V3.

Suppose that there is no proper subset I of { 1,.. ,n } such that ((g — U) U 6 / },s )has property (5). Then there is a p-morphism from T onto a descendant of Q.

Proof. Let T have the required properties. First there are t,/ < k such that we have for aU / < n : U{sV\ => Uj /Vi. For suppose that this is not the case. Take a Ui such that Ui Then for aU j < k there is an m < n, m / 1 with UjsVm. Hence

((ff - U ) U U{Kn|f« < n,m ^ l } , s ) has property (5).Now take i , j < k such that for all / < n : UisVi => Uj ßVi and take V/,VTO withUisV,,UjsVm. Let V 1 := V 2 := andv 3 := A ji>Uj Aüi} and define Tx - (f\,sx) by f x := (g - U ) U V/ U Vm and

*1 = « - |ji < «til ^ m,l})2u { (x i ,x 2)|xi € a ,x2 € V/, 3y € V 1 : xxsy}U {(x i t x2)|xj € a ,x2 € VmtBy € V 2 U V 3 : x jsy }

Now it can be checked that T\ is of type 6[{Vi,Vm}IU ) and therefore is a descendant of Q. Define a map p : T — ► T\ by p(x) = x for x € (g - U ) U VJ U Vm and p(x) € Vi if x € V 1 and p(x) € Vm if x 6 V 2 U V 3. Clearly we can define p in such a way that p[Vi) := {p(x)|x € V<} = Vj for Vj C V 1 and p[V<] = Vm for V j C V ' u V 2. It can be checked that p is a surjective p-morphism. H

92 5 SF-COMPLETENESS IN POLYMODAL LOGICLemma 5.4.12 Let 7 be a cluster valuation, Q = (y,r) be finite with root 0 and h = (h,<l). Let { bx C h\x € g} be a set with Vp[f>x|x 6 g] = h,bo ^ 0 and let a C g be downward closed in Q and b C h with p : hb 2; Qa so that (Qa — hi,) is recognized by { 6x|x € a}. Further let U € Cl{G),UC\a = 0, with Im c(U ) = {Uu ---,U k} C Cl{Ga). Define n := 7{G,U) and by := (J {6x|x € U}. Then h is subreducible onto an T € Fy*(G) if one of the following conditions is satisfied:

(Ext) There is no set c C h with GauU — hbuc so that (GauU — hbuc) is recognized by { 6*|x 6 aU(/}.

(R-Om) U is proper and for any V = i < k, there is a chain of reflexivepoints {y{ < ... < yj,} C by with V < y\ .

(Ir-Om) U is non-degenerated and for any V = p- 1(f/j),i < k, there is a chain of irreflexive points {y j <1 ... <1 yj,} C by with V < yj.

(Cl-Om) 6 ^ 0 and there are sets ci, . . . , c; C by, l > 1, with (c,, < ) ~ {U, r) and(a) Ci<j\ Cj for all i ^ j ,(b) for all i < k there is a j < l with p- 1(t^<) <3 Cj,(c) for all j <1 there is an i < k with p-1( ^ ) ^ Cj.

Proof. Define V{ := p-1((/,) for i < k.Suppose that (Ext) holds. Let Oi,.. .am C bu be a minimal set such that a,- n bx ^ 0 for all * < m, x € U and(a) for all * < k there is a j < m such that aj C 0 _ K,(b) for all i ^ j : Oaj fl a,- = 0,(c) for all j < m either (i) (aj, < ) ~ U or (ii) (aj, < ) a (w, < ) or (iii) (aj, < ) ~ (w, < ). The existence of such sets follows from the fact that Vg\bx\x € y] = h. Clearly m < k.

Case 1. m = 1. Then, by condition (Ext), is an infinite chain of reflexive or irreflexive points and U is non-degenerated. (i) Oi is a chain of irreflexive points. We proof that h is subreducible onto T — G[T*/U]. Take {yo < . . .< yn—1} Q a\ and define cx := bx for x € g — (a U U),cx := p- 1(x ) for x € 0 and Cj := {y , } for i < n. It is easy to check that V^[cr |x € f ] = h. Hence h is subreducible onto T . (ii) aj is a chain of reflexive points. Then, by condition (Ext), \U\ > 1. In this case h is subreducible onto G[T*/U].

Case 2. m > 1. Define a frame T = (f,s ) by / := (g - U) U U{ .. .U U'm with |t/j| = |{/|

s = r — (U x U)u { ( y i , y2)|yi e o, y2 € U-,P- 1( » 1) € Oa,}

u { ( y i , Jfti)|yi € U[, y2 € g - U, U r y2}

U { ( y i , y2)|yi, y2 € U(, V non-degenerated}.

Now define cx := p- 1(x ) for x € a, cx := bx for x € g -(a U U ) and let for t < m {cx|x 6 U-} be a partition of a,- with cx C Ocy for x, y € U( if U is non-degenerated. It can be checked

5.4 Tense Logics 93

that V?[cx\x £ f ] = h. It follows from Lemma 5.4.11 that there is a descendant of Q which is a p-morphic image of T . Hence h is subreducible onto a descendant of Q.

Suppose that (R-Om) holds. We show that h is subreducible onto 7 := G[T*/U]. Define d := {y%j\j < n,i < k} and let cx := bx for x £ g — (a U U),cx := p~l {x) if x £ a and C{ := {y £ d\dphd{y) = n — t — 1} for t < n - 1. Again it can be checked that v^ [cx|x e f ] = h.

Suppose that (Ir-Om) holds. Then the argumentation is as in (2), but reflexive points are replaced by irreflexive ones. This time h is subreducible onto G[T*/U].

Suppose that (Cl-Om) holds. We assume that ci,...,c/ is minimal with the required properties, especially that / < k. Define T = (/, s) by / := (g - U) u ci ... U ci and

« = r - (U x U)u {(2/1,2/2)12/1 < 2/2,2/1 > 2/2 € c,-,t < /}

U { ( 2/1 , 2/2) 12/1 ^ a ,y2 e Ci,p~l(yi) < y2}U { ( 2/1 ,2/2)12/1 € Ci, 2/2 € 9 - U, U t y2}

Define dx := p~l(x ) for x £ a, dx := bx for x £ g — (a U U), dx := {x } for x £ c\ U ... U c/. It can be checked that V?[dx\x £ f ] = h. Again, by Lemma 5.4.11, T is subreducible onto a descendant of Q. Hence h is subreducible onto a descendant of Q. H

Corollary 5.4.13 (Extension Lemma) For any cluster-valuation 7 , finite Q with root 0, h £ Fr(K4/Sf F^(Q)) and {bx C h\x £ g} with Vg[bx\x £ g] = h,60 ^ 0; If q : hb ~ Ga for a downward closed a C g is recognized by {bx\x £ a} then there are c D a andq : hc ^ Q such that (hc ^ Q) is recognized by {bx\x £ g}.

Proof, h is not subreducible onto an T £ F*(G). Hence the corollary follows by induction from Lemma 5.4.12 (Ext).

Corollary 5.4.14 For all cluster valuations 7 and finite Q with root 0 and all h £ Fr(K 4 /sf F *(G )): h is subreducible onto Q iff Q C h.

Proof. The direction from right to left is trivial. Now suppose that h is subreducible onto Q. Then there exists { 6X C h\x £ g} with Vg[bx\x £ g) - h and 60 # 0. Now the lemma follows from the Extension Lemma applied to a = b = 0. H

94 5 SF-COMPLETENESS IN POLYMODAL LOGICCorollary 5.4.15 Let J~ ,Q be finite and rooted frames. Then T is subreducible onto Q iff there is a Q\ 6 Fp(Q) with Q\ Q T.

Proof. The implication from right to left follows from Lemma 5.4.9. Now suppose that T is subreducible onto Q. Then, by Corollary 5.4.14, for all cluster valuations 7 there is a Q\ € F{Q) with Q\ C T . Take a 7 such that y(Gi,U) > \T\ for all Q\ and V 6 C7(£i). Then F 6)n{Gl\\g1\ <\F\} = FD(6) n {Silieil < |JF|}.H

5.4.2 Subframe logics above K 4 ( I V )

5.4 Tense Logics 95

Let us note the main consequences for subframe logics above A’4. The following theorem is essentially a reformulation of theorems known from F ine [85] and Zakharyaschev [92a].

Theorem 5.4.16 For all subframe logics A above K4 the following conditions are equivalent:

( 1) A is elementary.

(2) There is a cluster valuation 7 withFv {Frf (K4) - Frf {A )) C Frf (K4) - F rj(A ).

(3) There is a cluster valuation 7 such that for all finite sets F of rooted framesfrom A = A'4/^F follows A = K4/S F^(F).

(4) There is a cluster valuation 7 and a 7-closed set of finite rooted frames Fwith A = K4/S'F

If (4) is satisfied by 7 and F, then F r (A ) = {h € Fr(K4)\T %. h for any T € F }.

P roo f. ( 1) =*► (2). Suppose that there is no cluster valuation with the property required in (2). Then there is a finite frame Q A and a non-degenerated cluster U € Cl(G) such that (i) G[Tn/U] A for 0 < n € <*> or

(ii) U is proper and G[T*/U ] ^ A for 0 < n € <*>•But then all finite subframes of (7[(u>, < )/U ] are A-frames or all finite subframes of the frame £[(u>, < )/U ] are A-frames. Now, by Lemma 5.4.10, G is a p-morphic image of both of these frames. Hence F r (A ) does not have the finite embedding property. (2) =$■ (3) follows from Lemma 5.4.9 and (3) =$- (4) is trivial. For the implication from (4) to ( 1) let h £ F r (A). Then h is subreducible onto a G € F. Take a maximal T in (Fy(G), < 7) such that h is subreducible onto T . By Corollary 5.4.14, we have T C h. Hence A has the finite embedding property. H

Our work on variants and descendants begins to pay off with the following theorem.

Theorem 5.4.17 All elementary subframe logics above K4 are axiomatizable by a set T of formulas such that K4(<f>) is an elementary subframe logic for all <f> € T.

P roo f. Let A = K4fSiF be elementary. Clearly the theorem follows if there is a family A ;,t € /, of elementary, finitely axiomatizable subframe logics with A = V(A»|t € /). Define for G € F Ag := K4/Sf F^(G), with 7 such that F rj(K 4 ) — F r j{A ) is 7-closed. F~f(G) is finite and therefore Ag is a finitely axiomatizable and elementary subframe logic and A = M{Ag\G € F ). H

9 6 5 SF-COM PLETENESS IN POLYMODAL LOGIC5.4.3 P ro o f o f T heorem 5.4.2

P ro o f of Theorem 5.4.2. (5) O ( 6). The direction from left to right is clear. For the other direction note that F r (A ) = (F r (A + .t))+ which is an elementary class if F r (A + .t) is an

elementary class.(1) O (4) and (3) O (2) follow from the fact that K4.t is an iterated 5/-splitting of K.t by finite frames. Hence it remains to show (4) => (5) => (3) => (4).

(4) => (5). Suppose that A is not elementary. Then there is a finite rooted frame Q = (g, r) with G & F r (A ) and a non-degenerated cluster U G C l (G ) such that

(i) G [V J U ) |= A for all n > 0 or

(ii) U is proper and G[T*/U ] A for all n > 0.

We assume that there is no proper subframe of G with these properties. Let (i) be the case and |f/| = m. Define a tense frame H := (A, < + , A) by

h := u U {(0, n)|l < n < ra},

< := {(0, n)|l < n < m }2 U {(», (0, n))\i € w ,l < n < m} U € u,i < j } ,

and let A be the closure of

C := { a C h\a finite and (0,n) £ a for all 1 < n < m } U { {n j\ j G u>} U {((0, n))}|l < n < m}

under fl and —. It is standard to check that H is a descriptive tense frame. 7i looks as

follows: X *x -m-cluster0 1 2

By Lemma 5.4.5, G^W/U] = ((# — U ) U A,s+ ,s~ , B ) is a descriptive tense frame.

C laim 1 . G ^n /U ] ^ A+.t.Proof. Let U = { z i , .. .zm}. Define a set {bx\x G g } C B by bx := {a:} for x G g - U and bz% := {i j\ j G u;}u{(0, i ) } for 1 < i < m. It is easy to check that Vg[bx\x G g] = (g —U)Uh. Hence £ (a M V g - -,po)+ € A+.t.

Define 0 := Tfc(S/(0f[M /ff])).

C la im 2 . F r / (G ) C F r/ (A + .t).Proof. Let T = (/, s+ ,s “ ) G P r/ (0 ). Then T \= A + .t if any rooted generated subframe of = (/, s+ ) is a A-frame. Hence we may assume that is a rooted frame. Let z be the root of T . Clearly 0 is n-transitive for an n G w. Hence, by Corollary 3.1.11, there exists a set { 6x|a: e / } C B with Vyr[6x|z G /] = (g - V ) U h and b2 ^ 0. Now let

d := {x G /|6X n {(0 ,n )| l < n < m ) ^ 0}.

5.4 Tense Logics 97

Case 1 . d = 0. Then there is an n 6 w with (J {& rlx € / } C G^T'/U] £ F r ( A + .*). Hence

we get T £ Fr(A+.t).Case 2. d ^ 0. Then we have 6X C 0+6y and 6X C 0~by for all x ,y € d. Hence d is a subset o f a non-degenerated cluster V in T+. It follows immediately that there is

a cluster [/ ' in Q with U'rU and bx n U' ^ 0 for x £ V. Now it can be checked that

(g — U',r) is subreducible onto T * . Hence, by Lem m a 5.4.15, there is an T\ £ Fd (T* ) with C. (g - U',r). If T\ € F r / (A ) , then T € F r (A + .t). Suppose that T\ Frj(A ).

Then we have U n }\ = : Un ^ 0. Hence Fi[T*/U"] € F r (A ) for all n € a; which is a

contradiction to the assumption that Q is minimal.

C la im 3. F r ( 0 ) C F r (A + .t ) .Proof. Clearly Claim 3 follows from Claim 2 if all rooted frames in F r ( 0 ) + are finite. Let n £ u be maximal such that .there is a cluster V in G with |V | = n. Then let K := ( n + l , n + l x n + 1). Clearly we have (D ^ I V aj —► ->po)+ € © » (d ^ V / c —*- “,Po)~ £ 0 . Hence a frame in F r ( 0 ) + contains no cluster with cardinality > n, no strictly ascending

chains and no strictly descending chains. Further it is clear there is a A; £ a; with £ 0 . It follows immediately that any rooted T £ F r ( 0 ) + is finite.

Define A i := 0 f l A + X Then A i £ <SA'4.t,Ai C A + .t by Claim 1 and F r (A i ) = Fr(A+.t) by Claim 3. Hence A + .t is not strictly 5/-complete above A'4.t.

N ow let (ii) be the case and \U\ = m > 1 . Let K* be the reflexive closure o f H. It can be

checked that this is again a descriptive tense frame. Define 0 := Th(Sf(Gt[H*/U])). Now

the proof that A i := 0 n A.t satisfies A i £ SK4A, A i C A + .t,and F r ( A i ) = Fr(A+.t) is

completely analogous to the proof above.

Before going on with the p roof let us note that there is a crucial difference between the

form ula Vg and the form ula Vgt for a finite rooted frame G £ Fr(K4). This difference

reflects the fact there there are p-morphims p : H — ► G which are not t-p-m orphism s from

K* to Gl . W e sometimes write instead o f Vgt. Let us also note that a set {ax\x £ g} which recognizes (hf, ~ G) in Tt = (A, < , A) does not necessarily recognize (h[ ~ Gl ) in H *. The proof o f (5 ) => (3 ) is essentially a description o f how to m anipulate a set { a x |x £ g} which recognizes ( ht, ~ G) in H so that {a'x\x £ g} recognizes {h[ ~ Gl ) in H 1. It turns

out that such a manipulation is possible for H £ R fr ( K 4 / sf F * (G ) ) .

(5 ) =► (3 ) Let A = Ar4/5^F , F a set o f finite rooted frames, and let 7 be a cluster valuation

with F y (F ) C F . W e p roof by inverse induction on < -, that the following holds for all 7 -closed sets F i C F :

( i ) F { defines an iterated ^/-splitting o f K4.t

(ii ) (K4/sfF1).t = K4A^F\

Then we have for F i := F that F { defines an iterated Sf- split ting o f K4.t with

A = K4/SfF\.

Let G £ F with root 0 and suppose that ( i ) and (ii ) hold for F^(G)- N ow define

9 8 5 SF-COM PLETENESS IN POLYMODAL LOGIC0 := K4/Sf F^(G). 0 and 0 + .t are elementary subframe logics. Hence they are R- persistent and Rfr(0+ .t) = {H £ Rfr(K4.t)\Ti % A for all T £ F * (G ) } - Q /S*G is elementary and therefore Ä-persistent and we have

R fr {e /S*G) = { H £ Rfr(B)\G % A}.It foDows for all h £ F r (0 + .t) : Apo Is consistent in h if and only if G Q A. Hence(i) and (ii) are shown for F^(G) if

(Rec) For all H £ Ä/r(0+ .t) : If Gl E A, then there is a set {bx\x £ g } C A with

Vg[bx\x € g] = A and b0 ^ 0.

Let = (A, < + , € Ä/r(@+ -*) and let £* C A. We will often write < instead of<J + .

Claim 1. Let a C p b e d o w n w a rd closed an d let {bx\x £ <7} C A b e a set w ith

( + ) There is a c C h such that hc ~ Gl and so that {bx\x £ y } satisfies (R D ),(R £) and (R n ) for (hc ~ Gl) in H.

Then there is a set {ax\x £ <7} C A with

(i) if V £ Cl(Ga) is non-degenerated and d C A with p : ~ G such that (A j ~ G)is recognized by {ax\x £ g } in W+ , then(1.1) there is no y £ (J {ax|a: € U} with p“ 1 (f7) < y and yiJ y,(1.2 ) if U is proper then there is no y £ |J{ax|a; € 17} with p~l (U) < y and

y4p~l (U)-

(ii) «X C bx for all x £ g.

(iii) Vg[ax\x £ g] = A and ao # 0.

(iv) Vx £ a, y £ ax there is a 6 C A with y £ 6 such that Aj}- ~ (7a and (A+ ~ £a) isrecognized by {a~\z £ a } in .

Proof. The proof is by induction on \Cl{Ga)\ for downward closed a C g.Let a = 0 and let {bx\x £ g } C A be a set with property (+ ) . We prove by induction on \Cl(F)\ for all generated subframes T of G: There is a set {ax\x £ g } C A with ax C bx for x £ y, ( + ) and

(a ) Vjr[ax\x £ f ] = A.

(b ) For all {7 £ C7(.F) and <f C A with q : ~ G and q(y) = a: iff y £ ax there

is no y £ |J{ax|a; £ U } with q~x{U ) < y and y J y. If U is proper then there is no

y € LKax|s € tf} with r l (*0 < y and y$

5.4 Tease Logics 99

Suppose that {ax\x £ g} C A has these properties fo r T. Let U be a cluster in Q such

that all strict r + -successors o f U are in / .

Case 1 . U is degenerated. Then define dx : = ax fo r x £ g — U and

dx : = ax fl n { 0 + a j * r y }

fo r { z } = U. It is easy to check that {dx\x £ g) has all the required properties for

(/ U 17, r ) . (F o r ( + ) note that for {x} - U we have 0 + a x fl ax = 0 by (R □ ) . )

Case 2. U is non-degenerated. Define n : = 7 (G, U). N o w take a set di C h and qx : / i j ~ Q with qi{y) = x iff y £ ax for which there is a m axim al chain

b = {y\ <3 + . . . <3 + ym} Q U { ax k € U}o f irrefiexive points with y <1 + y\ for a y £ qil{U). B y Lem m a 5.4.12 (I r -O m ), we have

m < 11. I f b = 0 define a\ : = ax for x £ g . O therw ise choose a cycle free fram e AJ with

(AJ < - hijub) for a z £ q^ (U ) and

• x <\fcy\i x < y and x £ b U U, y £ b.

N o w choose a set {cy\y £ 6 U U} which recognizes (AJ <z jV) in with

• cq- ' ( x) Q &X for x € U .

• cy C (J {a x |z € U} for all y £ b U U.

Define a\ := ax for x £ g — U and a\ := c ^ -i^ for x £ U. {a\\x £ g} still has (+ ) and

the required properties for T . But now we have for all V £ C l ( ( f U U,r)) and d C h with q : li$ ~ G and q(y) = x iff y £ alx that there is no y £ (J{a*|x € V } Wlt^ < Vand y<fl y: Assume to the contrary that there is a Q h, an isomorphism q2 and a y with these properties. Then <72(2/) = x iff y £ ax because a* C ax for x £ g. Further we have y £ cg- i ^ for an x £ U. Hence there is a chain { j/j <J+ ... <J+ ym} C (J{ax|x £ U}

with y <]+ y\. But then q j1 (f7) <J + y <J + yi <j + ... <j + ym, which contradicts the assumption that d\ was chosen so that m is maximal.If U is proper we still have to eliminate reflexive chains. The construction is analogous: Take a set d C h such that q : ~ G with q{y) = x iff y £ alx for which there is amaximal chain b — { 3/1 <j + ... <j + ym} C {a].\x £ U } of reflexive points with y < yi for a y £ q~l (U ). By Lemma 5.4.12 (R-Om), we have m < n. If 17 was not proper or b = 0 define ax := alx for x £ g . Otherwise choose a cycle free frame AC with (AC < z h ju6) for a z £ q~l (U ) and

• x </cy if x < y and x £ bUUy y £b and x ^ y.

A ga in choose a set { cy\y £b\JU} which recognizes (AJ <z h£uU) in 7t+ w ith

100 5 SF-COMPLETENESS IN POLYMODAL LOGIC• c , - i (x) C <4 for x € U.

• cy C U{<4|x € f/} for any y € 6 U U.

Then define a2x := a], for x € g - U and a2 := c9-i(x) for x € U. Again {<4|x € 5} still has (+ ) and the required properties for F and we have for all proper V £ C l({f U U,r)) and dCh with q : h j ü Q and q(y) = x iff y € a2 that there is no y £ Ufaxl1 € V'} with q- , (r/) < y and q~1 (U). This is proved as above (by using that the irreflexive chains are eliminated). Now define for x £ U a set cx by

cx := a2 n n{0+a2|xry}.

Define dx := a2 for x £ g - U and define for an xi £ U:

dXl := cXl n □+(*) D {cVl -» O+Cjolin,^ € U).

For all y e U with y ± x\ define dy := cy fl O-dxx. Then {dx\x e g} has (-f) and all the required properties for (/ U r/, r).

It follows for T = £7 that there is a set {ax\x € <7} C j4 with (i),(ii) and (iii). The condition (iv) is trivial for a = 0.

Let a C g be downward closed and suppose that Claim 1 is shown for a. Let U € Cl(G) be such that Uf\a = 0, {I7 i,.. .17*} = Img(U) C a,and suppose that { 6x|x € <7} C A satisfies (-f). Define n := *y(Gy U) if U is non-degenerated. We construct sets {ax\x € g}yi < 10, and finally get from {a*°|x € g} a set {dx\x € g} which will have the required properties for flU(/. We will frequently use the following principle which easily follows from the Extension Lemma:

Extension Principle: Let {bx\x e g } C Ayi = 1,2, be sets with b\ C 6* for x € g and Vg[b]c\x e g] = h. Let p : li£ ~ Qa be recognized by {b}.\x € a} in 7f+ . Then there are cD b and p : h+ ~ Q such that (h+ ~ Q) is recognized by { 6*|x € g} in 7f + .

Claim 1.1 (C 1) There is a set {a^|x € g} C A with ax C bx for x € <7 and properties (i) ... (iv) for a such that( 1) if U is non-degenerated then there is no chain of irreflexive points

{yi < + . . . < + y„) C U{Oxl* € t/},(2) if V is degenerated then x^j y for all x,y £ LKa*lx € f/}.

Proof of Claim 1.1. By induction hypothesis, there is a set {a x|x £ </} C A which satisfies (i) ... (iv) for a. Clearly {ox|x £ 5} satisfies (C l) if U is degenerated. Now suppose that U in non-degenerated and that (C l) does not hold for {a x|x € g}. Then take a dt C h and 0i : - Q with (&£ ~ Q) recognized by {ox|x € g) in H+ such that there are chains{yj < + . . . < + y^ ,} C {ax|x€ U) of irreflexive points with q i1(Ui)<i + y\ fort < Jb so that

5.4 Tease Logics 101N := ]£(mtn(»,ro,-)l* < k) is maximal.

We have N < kn, by Lemma 5.4.12 (Ir-Om). Define := {yj|t < k ,j < m,-} and let

b := by U \J{qT\Ui)\i < k} U q ^ (U ). Clearly q ^ (U ) D fej = 0.

• x <*; y if x < y and x € b - g, l (U ),y e h U q1 1 (U ).

(Note that K. is not rooted. So, JC < h+ is not defined so far; but the meaning should be clear.) Then choose a set {cx|x € 6} C A which recognizes (AC < /ijj") in 'H+ and

• cy C {o x|x € U} for y €

* c q~l (x) - °x f° r x € U Im g(U ) U t/.

Now define a* := ax for x € S — (U Im c(U ) U U ) and «4 := c9- i ^ for x € (J Im g(U ) U V. Clearly, {a x|x € g} has property (+ ). Hence, by induction hypothesis, there is a set {<4|x € g} C A with ax C ax and (i) . . . (iv ) for a.Define := <4 n f ) { 0 _ ay|y € |J for x € U and ax := a£ for x € <7 - l/.

We check (C l). It follows immediately from Lemma 5.4.12 (Ext) that {a£|x 6 g} still satisfies (i) . . . (iv ) for a. Suppose that there is a chain of irreflexive points

{lb < + . . . < + yn} C € U).

102 5 SF-COMPLETENESS IN POLYMODAL LOGICIt follows from Lemma 5.4.12(Ext) that i>i / 0. Let yi £ a%. Now choose a {/,- € Img(U) with m, < n and take a y £ ax with x £ Ui and yi y. By condition (iv), there are a set c C h with y € c and p : hf ~ Qa recognized by {o^|i € a }. By the extension principle there is a set c C cj and p : hft ~ Q which is recognized by {ax\x £ g} in H+. For any p_ 1 ( Uj) there is a chain of irreflexive points

the assumption that q\, d\ were chosen so that N is maximal.

Claim 1.2 (C2) There is a set {a® |x € p } C A with ax D ax and property (C l) such for all d C h with q : /ij ~ Q and (/ij ~ Q) recognized by {a®|x £ g} there is no y £ (J{ax|x £ U} with y < g- 1(r/) and y<fi y.

Proof of Claim 1.2. The claim is trivial if U is degenerated. Otherwise choose d\ C h and q\ : ~ Q recognized by {ax\x £ p } in 7i+ such that there is a maximal chain of

irreflexive points bj = {y i < + ... < + J/TO} Q LK®*!1 € U) with ym < + q^ iU ). We have m < n, by (C l). Define 6 := &i U q^l (U).

Now choose a cycle free frame fC <z (!) with

• x < icy if x < “ y and x £b,y £b\.

Choose a set {cy|y £ 6} C A which recognizes {K. <- h^) in H~ with

• Cy C \J{ax\x £ U }

• C ax for x £ U .

Define a* := c9- i (r) for x £ U and := a% for x £ g - U. Clearly {<z*|x £ g) still has

5.4 Tease Logics 103property (+ ). Hence, by induction hypothesis there is a set {o®|x £ <7} with a* D a® and (i) . .. (vi) for a. It follows with the extension principle that {a®|x € g } has property (C2).

Claim 1.3 (C3) There is a set {a®|x € g} C A with a® C a® and property (C2) such that there is no chain of reflexive points {j/i < + ... < + j/2n} Q U i “ *!1 € U}, if U is proper.

The proof is analogous to the proof of Claim 1.1. This time one has to use Lemma 5.4.12 (R-Om) and property (C2) of {o®|x £ <?}• Since we did not exclude chains of length < n of irreflexive points in (C l) we get only the bound 2» for the reflexive chains.

Claim 1.4 (C4) There is a set {a£|x £ g} C A with a7x C a® and property (C3) so that if U is proper then for all d C h with q : /ij" ~ G and (/ij ~ G) recognized by {a£|x £ g] there is no y € U {a*lx € U} with y < q~l (U ) and y.

The proof is analogous to the proof of Claim 1.2.

We are now ready to finish the induction step.

Choose d\ C h and <71 : /ij- ~ G recognized by {ax|x £ g} in 7 such that there is a maximal set I C {1 , . . . , k} with the following property:There are sets V j, . . . V/ C (J{oJ|x € U} with {Vi, < ) ~ (U, r) and

(a) Vi<fi Vj for t ^ j,

(b ) for all i £ I there is a j < l with q^ iU i) <J + Vj,

(c) for all j < l there is an i < k with Vj.

We may assume / < k and have, by Lemma 5.4.12 (Cl-Om), I ^ { 1,.. .k}.Define := Vi U ... U V/ and 6 := 61 U 9 f1(U Img{U) U U). Clearly 61 D = 0. Thefollowing picture shows an idealized version of the situation.

1 0 4 5 SF -C O M P LE TE N E SS IN PO LYM O D AL LOGICChoose a cycle free frame 1C < with x < % y if x < y for x £ l ( \ J € 6i-Take a set {cy\y £ b} C A which recognizes (K < h£) in with

• Cy C U { al\x £ U ] for y £ 6i

• cg- i ( x) C a7x for x £ U U U Im g (U ).

Case 1 . U is degenerated. Define a| := a7x for x £ g - (U U Im g {U ) ) and a® := c -i

else. {a||x G <7} has (4-). Hence, by induction hypothesis, there is a set {ax\x £ g } with

ax C a% and (i) . . . (iv ) for a. Define dx := a?x D if x £ U and dx := a®,else. It is easy to check that {dx\x £ g } satisfies (i) . . . (iv ) for a. We check (iv ) for aU U. Let y £ ax with { x } = U. It is clear that y is irreflexive. Take a j £ { l , . . . f c } - /. By

definition there exists a z £ { } {d x\x £ U j } with z < + y. By (iv ) for a, there are a set d C h with z £ d and p : /ij ~ G such that (/ij ~ Q) is recognized by {dx\x £ g } in W + . By

definition there exist sets V j , . . . , C \J{al\x £ U } of degenerated cluster which satisfy (a ), (b ), (c ) for / and p. By the Extension Lemma, there is a c D d such that p : h+ ~ G is recognized by {a £\x £ g } in H + . Hence we get p~l (Ui) < + y for all i < fc, because otherwise {j/}, . . . , V! would satisfy the conditions (a ), (b ), (c ) for I U { j } which wouldcontradict the assumption that / is maximal. It follows that {dx\x £ y } satisfies (iv ) for a U U .

Case 2. U is simple. Let V{ = {z t} for i < L By (C l) the sets c~,z £ b\ contain no chains of irreflexive points of length > n. It follows from Lemma 4.1.2 that there exist cfs C c2,c'z £ A, with z £ c' for z £ 6i and

c'z — O+c'2 = h.

Let { y } = U . Now define a% := cq- i (xj fl f| { <>+cz k r 1( :c) < z, z € M for x £ \JIm g (U ) and a® := c9- i^ and a® := a£, else. Clearly {a®|x £ g } satisfies (-f). Hence, by induction hypothesis, there is a set {a*|x £ g } with a?x C a% and (i),..., (iv) for a. Define

a\° := H Pi{0 -al\xryyx £ a} and then

dy := a™ fl □ "(1)(aJ° -> O"aJ0)

and dx := a® for x € g - U. It follows from (C2) that { dx\x £ g} still has (i) ... (iv) for a. We now check (iv) for a U {y}. Let z £ dy. Then there is a reflexive point z^£dy with z\ < + z. It follows as in the proof above that there are a d C h and p : hJ ~ Q recognized by {dx\x £ a} with p" 1 (£/*•) <1 z\ for all t < k. From (i.l) we get that z is reflexive and p~l (Ui) <J z for all i < k.

Case 3. V is proper. The sets \Jicz\z € VJ},t < fc, do not contain chains of length more than 3n. Hence there are sets c' C cz, c' € A, with z £ c' for z £ bi with c'Zl 0 +c' 2 = h

5.4 Tense Logics 105

for zj,Z2 £ Vi,i < /. Now the construction is analogous to the construction above. Define a* for x € U Img(U) as above and a| := cq- l(x) for x £ £/ and a® := a£, else. Again there is

a set {c£|x £ g} with a% C a| and (i)v .. (iv) for a. Define aj° := a9 C \ { ^ ax\xry’>x € a} for y £ (/ and then take a y £ (/ and let

:= aj° n c H 1» n {a j° - 0~4°|z 6 U}

Now define dx := a™ fi O+dy for x £ U - {y } and dx := a9 for x £ g - U. It follows from (C 4) that {dx\x £ g} still has (i) ... (iv) for a. We check (iv) for aUU. Let z £ dx,x £ U. Then there is a set V with V < + z, (V, <! + ) ~ U and V H dy ^ 0 for y £ (/, because U{^y|j/ € F } contains no chains of length > Sji. As in the proof above there are d C h and p : hJ ~ <7 recognized by { d x |x £ <7} with p- 1 (F t) <1 V for i < k. By (i.2 ) we get that z < + V and therefore there is a cluster V D V with 2 £ V 7. (iv) follows immediately. Claim 1 is shown.

Proof of (Rec). Let hc ~ Ql for a c C h. By Lemma 3.2.5, there is a set { 6x|x £ g} C A with (R £ ), (R H ) and (R □ ) . Hence, by Claim 1 for a = g, there is a set {a x|x £ g) C A with (iii) and (iv). We show V^[ax|x £ g] = h. The interesting step is ax 0 “ ay = h for yrx. Let z £ ax. Then, by (iv), there is a b C h with z £ 6 and h* ~ Q recognized by {ax|x £ g}. It follows that there is a z\ £ ay with z\ <J z. Hence z £ 0 “ ay.

(3) (4) Let A + .t be an iterated S/-splitting of K 4 .t by finite frames. We show that A iselementary, from which follows that A+ .t complete. Then strict Sf-completeness follows from completeness. Suppose that A is not elementary. It follows from the proof of (4)(5) that there is a logic 0 £ S K 4 .t with 0 C A+ .t and F r (0 ) = F r(A + .t). But then A+ .t is not an iterated 5/-splitting of K 4 .t by finite frames. We have a contradiction^

Let us note the following corollary to the proof of (4) => (5).

CoroUary 5.4.18 Let A £ SK4.t. Then A is not strictly Sf-complete above K4.t if G C A+ C G .3 and A is not strictly Sf-complete above S4.t if Grz C A+ C Grz.3.

Proof. Let G C A+ C G.3. Then Q = [• ] A. If A is not complete there is nothing to show. Otherwise define 7i as in the proof of (4) => (5) for m = 1 . Then (7*[?f/F] = H where the cluster U should be clear. Define 0 := Th(Sf{H)). It follows from the proof of(4) =* (5) that the rooted 0-frames are exactly the frames in F := {T*\n £ u;}*. Now A is complete and therefore F C F r(A ). Hence we get 0 n A C A but F r (0 n A ) = F r(A ). Now let Grz C A + C Grz.3 and suppose that A is complete. Then Q = | — A. Define H* as in the proof of (4) =► (5) for m = 2. Then the rooted frames of 0 := Th(Sf(H*)) are exactly the frames in F # := {7£|n £ u;}*. Again 0 n A is a subframe logic with the same Kripke frames as A but 0 fi A C A. H

106 5 S F -C O M P L E T E N E S S IN P O L Y M O D A L L O G IC

5.4.4 Some Remarks on the FM P

We now use the techniques of the last chapter to deduce some negative results concerning the fmp of tense logics.

Theorem 5.4.19 Let A be a subframe logic abot>e KA and let Q £ F r (A ) be finite such that Clu{Q)x = (C l(Q ),r)x is a transitive tree. Then A+.t does not have the fmp if there is a non-degenerated cluster U € Cl{Q) such that (i) G[T*/U] € F r (A ) for all n € w or (ii) U is proper and G[T*/U] € F r (A) for all n € u.

Proo f. Let A ,Q = {g,r) and U € Cl(Q) satisfy (i). Define H := C7*[(u;,> , <>/f/].Claim 1. "H (= A+.t.We show H+ f= A. Suppose that H+ is subreducible onto T € Frj(K A ) with T rooted. Let { 6X C h\x € / } be a set with Vg[bx\x £ f ] = h and to ^ 0- Then define

ax := {y € bx\Vz €bx :y < n+ z => z < n+ y} for x € /.Now H contains no infinite strictly ascending < -chains. Hence V^-[ax|x e /] = h and flo 5 0 and (J {ax|x € / } is finite. Hence there is an n € u such that QY^n/C] is subreducible onto T . Thus, T € F r (A) and Claim 1 is shown.Reserve a variable px for any x € g. Now define

<t> := V (p.r-)[Px|a: € tf]A V (ff_ I/ir)[px|x e g - u ]

A A <Pv -*• 0 +px\y e U,x e g - U, yrx)

A f$Pv -* 0 +P*|y e g - U , x e U , yrx)

It should be clear that A po is consistent in Ti and therefore that □ <f> —► ->po £A+.t. We show that this formula holds on any finite A+ .t-frame. Let T — {f,s+ ,s~) be a finite A+ .t-frame and suppose that there is a set { 6X C f\x € g} and y € / with

y € A po[6x|x € <?]• Take an Xj G g with dpg{x$ = 0 and Ur+x\. Now take ay\ € bx, with ys+yi. Clu(Q)x is a tree. Hence there is a set 6 C TrJ-_ (j/i) so that

p : hi ~ {{z € jf|zrx i},r+ , r - ) with p{z\) = z iff z\ € b2.But then it follows again from the condition that Clu{Q)x is a tree that there is a set c D 6 with T c ~ Q. It is shown that T & Fr(A+ .t).

The case (ii) can be proved analogously. In this case one has to consider the frameG[(u>,>,<)/?/]• H

Exam ples. Let A € SKA. Then A+,t does not have the fmp if

( 1 ) G C A C G. 3 or

(2) Grz C A C Grz. 3 or

(3) KA.Z C A and A is not elementary.

5.4 Tense Logics 107

Proof. (1) In this case the conditions of Theorem 5.4.19 are satisfied by Q = [T ] . (2) In

this case the conditions of Theorem 5.4.19 are satisfied by Q = | •—— |. (3) Let A D A'4.3 be non-elementary. By the characterization of non-elementary subframe logics above A'4 there exists a finite and rooted Q £ Fr(K4.3),G & F r (A), so that (i) or (ii) is satisfied. Clearly Clu(G)x is a tree.

Corollary 5 .4.20 Let A\ be a subframe logic above A'4 with the properties of Thm. 5.4-19 and let A2 be an elementary subframe logic above A'4 such that A := A j’.A.J.t is a conservative extension of A\. Then A does not have the fmp.

Proof. We have only to check that the relevant frames 7i\ = Gl[(u, > , < )/U ] respectively Hi - </*[(<*>, > , <)/(/] are Ajf.A.J.t-frames. We prove this for 7i\. We know from the proof of Theorem 5.4.19 that TC\ is a Ai-frame. TCf is a A2-frame if any finite subframe of is a Ai-frame. But the finite subframes are the subframes of frames in F := {G[T£/U]“ \n £ u;}. It should be clear that these frames are A2-frames if A+.A.J.t is a conservative extension of A l H

Examples. Let A £ S K A satisfy (1),(2) or (3). Then A + ./1“ .t does not have the fmp.

Note that the elementarity of A 2 is essential in Corollary 5.4.20. Take for instance G .3+.G .3~ .t. This logic has the fmp (see the following theorem) and is a conservative extension of G.3.

Let us mention the following positive results concerning the fmp of tense logics. They will be proved elsewhere since the proof methods are different from those introduced in this paper.

Theorem 5.4.21 (1) If A is a monomodal and elementary subframe logic above A'4, then A+ .t has the fmp.(2) All R-persistent subframe logics above K4.I+.I~.t for an n £ lj have the fmp.(3) All R-persistent subframe logics above K4.I*.t have the fmp.(4) All subframe logics above G*.G~.1+.I~.t have the fmp.(5) All subframe logics above G.3*.G~.t have the fmp.

The proof of this theorem uses an extended version of the elimination theorem of Fine [85]. The main difficulty is that one has to take maximal points in both directions, < + as well as We conjecture that the converse of Theorem 5.4.21 (1) is also true. Notice that for all non elementary subframe logics A above A'4 there is a finite and rooted frame G Fr (A) which satisfies (i) or (ii) of Theorem 5.4.19. But there is not necessarily a frame G with these properties so that Clu(G)x is a transitive tree, and in this case the proof of Theorem 5.4.19 does not work.

The following theorem, which will be proved elsewhere too, shows the limits as regards positive results concerning the fmp for A-persistent subframe logics above K4.t.

108 5 SF-COMPLETENESS IN POLYMODAL LOGIC

Theorem 5.4.22 There is an R-persistent subframe logic above -*®2 wiJ/iotiJthe fmp.

A+.t is an iterated 5/-splitting of K.t if A is an elementary subframe logic above A'4. Hence we get from Theorem 5.4.21 ( 1):

Corollary 5.4.23 Let A be a finitely axiomatizable and elementary subframe logic above K 4. Then K.t(<f>sf) = A+ .* is decidable.

6 Splittings and ^/-splittings in some sublattices of A'2

109

So far we have investigated splittings mainly to decide whether a given logic is an iterated ^/-splitting. But there is another point of view concerning splittings: Given a logic A, then characterize the finite rooted frames which (Sf-) split £A. Corollary 2.4.15 solves this problem for logics which are m-transitive for an m 6 w. In Chapter 3 we have given a characterization of the finite rooted frames which split SAfn and S((#nT ) and got the interesting corollary that a finite rooted frame .9/-splits Afn if and only if it splits A/*„, but that there exist frames which .9/-split C*OnT but do not split $ nT. We will now give solutions to this problem for the lattices £(K4 (/) K4)^ £A'2.f, £(.95 OjO .95). For some of the proofs in this chapter the reader should have at hand Kracht & W olter [91].

For a 2-frame (p, <31 , <12) ( (0>ri>r2)) let < := U < 2 (r := n U r2).

Definition 6.0.24 A K 4 C>0 K 4-frame Q contains no internal cycles if for all n > 1 andb — {z i < X2 < ... < xn} C g with x\ = xn there is an i 6 {1 ,2 } with b2 n <Jt- = 0.

Define by induction (D )1! := □ 1± , (D )2n+1± := □ 1(D )2n_Land (D )2n+2_L := □ 2(D )2n+1_L.

Theorem 6.0.25 Let Q be a finite and rooted K 4 Qjo K4-framc. Then the following conditions are equivalent:

(1) G contains no internal cycles.

(2) There is an n with Q (= ( □ )WX.

(3) Q splits K4 ® K4.

(4) G Sf-splits K4 ® K4.

Proof. Let Q = (^, r j , r2) and let 0 be a root of Q. The equivalence of (1) and (2) is easy to check. (1) ^ (3). Suppose that Q contains no internal cycles. Let m := 2\g\ -f 1 and let Tt — (A, <h, < 2, j4) € Rfr(K4 ® K4) with H,ß,x o (m)A$ A po- We show that TV™+1(x ) — Tr™(x) = 0. Then we have 7i,ß,x O ^ A ^ p o and it follows from the Splitting Theorem that Q splits £(K4$iK4). Suppose that TV™+1(x ) — Tr™(x) ^ 0. Then

we may assume that there is a set 6 = {a; < 2 * i < 1 *2 < 2 *3 • •. < 2 *m } Q h with x, ^ Xj and x ^ X{ for all * ^ j . We have K,ß,x [= cKm) \f(pz\z € g). It follows immediately from(1) and the definition of A$ that |6*| < 2 for z € g and bz := ß(pz) n 6. Hence |6| < m, which is a contradiction. The implication from (3) to (4) is clear. It remains to show (4) ^ (1). Clearly we have

K4 = Th{{Q € Frj(K4)\G has a root x such that {p|p < x} = 0 }).

It follows immediately from the proof of the completeness theorem in Kracht & W olter [91] that

A4 $ A4 = Th({G € Frj(KA ® KA)\Q is rooted and contains no internal cycles }).

110 6 SPLITTINGS AND SF-SPLITTINGS IN SOME SUBLATTICES OF fif2

Now consider a finite and rooted A'4® K 4-frame T with an internal cycle. It follows from the equivalence of ( 1) and (2) that T £ Fr(S f(G )) for all finite A 4® A4-frames Q without internal cycles. Hence Th(Sf(G )) is not prime in S(KA ® A'4) and therefore does not Sf-split A'4 ® A4. H

One might hope that K4.t is an iterated splitting, or at least an iterated S/-splitting, of A 4 ® A'4 by finite frames. This hope is destroyed by the following frame. Define PL = (h, by h = u + 2,

<1 = { (^ ,w ) }u { ( i , i )|t , j€w ,i> i } ,< 2 = {(u;,w)}U {(w + l ,w ) } U {{ i,j)\ i,j € w ,i < j } ,A = {a C h\u G a and a cofinite or u> £ a and a finite }.

It is readily checked that A := Th(S f(H )) 3 A'4 ® A4 and that Fr(Sf(PL)) C Fr(K4.t). It follows immediately that F r (A fl K 4 .t) = F r (K 4.t) and that A fl K4.t C K4.t. Hence K4.t is not strictly 5/-complete above A4 ® A4 and is not an iterated 5/-splitting of A'4 ® A'4 by finite frames.

Splittings of £K 2-t are characterized in K racht [92]:

Theorem 6.0.26 ( {0 }, (0,0)) is the only frame which splits K2.t.

Defin ition 6.0.27 A tense-frame G = (<7»< li»< l2) t-cycle free if(i) (g, 3 and {®o < <J . . . <J xn} C g with Xo = xn we have Xi — xn_i

or xt_ j = x,+1 for an i with 0 < i < n.

For a bimodal frame G with root x let Gl denote the minimal tense frame with G <x Gt ■ (Here, minimal means minimal with repect to < x).

Theorem 6.0.28 For a finite and rooted tense frame G the following conditions are equivalent:

( 1) There is a 2-tree T such that G = T*.

(2) G is t-cycle-free.

(3) G Sf-splits K2.t.

P roo f. The implication from ( 1) to (3) follows from Theorem 5.2.1. (3) =S> (2). A simple variant of the ramification method shows

K2.t = Th({G € Fr/(K 2.t)\G rooted and t-cycle free } ) .

I l l

Now let T — (/ ,ri, 1*2) be a finite and rooted not f-cycle free frame. We show that T £ Fr(Sf(Q )) for all finite f-cycle free frames Q. Then Th(Sf(!F)) is not prime in 5AV* and therefore does not 5/-spDt K2.t. So, assume to the contrary that there is a finite t- cycle free frame Q = (p, < 1 , < 2) and a surjective p-morphism p : Q — ► T. Clearly (/ ,t*i ) and (/, 7*2) are cycle free. Hence there are n > 3 and { x0r x i r . . . r i „ } C / with x0 = xn such that x\ ^ xn_i and Xj_ 1 ^ Xj+i for all j with 0 < j < n. It foDows that there is a sequence (Zm|m € a;) with zm £ g and zm < zm+1 and p(zmn+t) = xt for 7/1 € a; and 0 1 with zm_ 1 = zm+i- It foDows that p(zm_ 1 ) = p(zm+1 ), which is a contradiction.(2) => (1). Let Q = (p,**!, 7*2) satisfy condition (2) and let x be a root of Q. Clearly there is a 2-tree T with root x such that T <x Q. Suppose that T* ^ Q. Let T* = (p, < 1 , < 2)- Then there exist 7* < 3 and {x 0 < x2 < .. .xn- i } C p with x:- ^ Xj for t ^ j but x0rxn- i . But then Q is not f-cycle free. H

It follows with Theorem 5.2.1 that all union-5/-spDt tings of K2.t are A-persistent.

Definition 6.0.29 An 55® Sb-frame Q = (p, = { * } .(ii) For all n > 3 cmd xo < x i < .. . < x n with xq = x n so that there is an i with

Xi<fl x 0 there are ji, j 2 with j\ ^ j 2 and 0 < j i , j 2 < n such that Xjx = x ^ .

For a 2-frame Q with root x let Qss denote the minimal 55 ® 55-frame with £ < x Qss.

Theorem 6.0.30 For a finite and rooted 55®55-/rame the following conditions are equivalent:

(1) There is a 2-tree T with T ss = Q.

(2) Q is S5-cycle free.

(3) Q Sf-splits 55 ® 55.

Proof. The implication from (1) to (3) foDows from Theorem 5.2.1. (1) =» (2). It foDows immediately from the proof of the completeness theorem in Kracht & WOLTER [91] that

55® 55 = Th({Q e F r/ (55® 55)|£ rooted and 55-cycle free } ).

Now let T = (/ ,r i ,r2) be a finite and rooted not 55-cycle free frame. We show that T £ Fr (Sf(Q )) for any finite 55-cycle free frame Q. Then Th {S f (T )) is not prime in 5(55 0 55). Assume to the contrary that there is a finite 55-cycle free frame 6 = (p, 3 and { x 0r x i r . . .r x n } C / with x0 = x n and x t / x 0 for an i < 71

112 6 SPLITTINGS AND SF-SPLITTINGS IN SOME SUBLATTICES OF Af2

and i j , ^ Xj2 for / j 2 and 0 < j\ ,j2 < «•Then there is a sequence (zm\m £ oj) in g with zm < zm+j, p(znm+t) = x, for to € oj

and 0 < t < n. Now p is finite and therefore there exist thi, to2 with Toi ^ m2 and zm, = zmj. We may assume that M := |toi — m2| is minimal with this property, p is a p-morphism and Q is 55-cycle free and therefore M < n. But then p(zm ,) = p(z,„2) leads to a contradiction.Case 2. There exist xo,xi £ / with xo / xj and xorjXi and xor2xi.Then there is a sequence {zm\m £ oj) with z2m < i z2m+1, p{z2m ) = * 0, 2m+1 <J2 z2m+2 and p(z2m+1) = xi for m £ oj. There are i , j £ u with t ^ j and z, = z,. We assume that M := \i — j\ is minimal with this property. Then, by condition (i) for 55-cycle free frames, M > 2 and, by condition (ii) for 55-cycle free frames, M < 3. A contradiction. It remains to show (2) =>• (1). Let Q = (p ,r i,r2) satisfy (2) and let x be a root of Q. Clearly there is a 2-tree T with root x such that T <x Q. Suppose that T Sh ^ Q. Let T Sh = {g, <Ji, < 2). We may assume that there exists a path {xo <J xi <1 ... < xn_ j } C g with n > 3 and x, ^ Xj for i ^ j such that xo5iiXn_i but xorixn_ j. Then, by condition (i) for 55-cycle free frames, xo f 2xn-i ■ Clearly this is a contradiction to condition (ii) for 55-cycle free frames.H

It follows from Theorem 5.2.1 that all union 5/-splittings of 55® 55 are 5 -persistent.

For 1 < a < oj define 55(a) := Th({a,a x a )). Then it is easy to generalize Theorem 6.0.30.

Theorem 6.0.31 For 1 < a, ß < oj and finite rooted 55(a) ® S5(ß)-frame$ Q the conditions ( 1) and (2) are equivalent:

( 1) Q is S5-cycle-free.

(2) Q Sf-splits 55(a) ® S5(ß).

Again there is a crucial difference between 5/-splittings and splittings:

Theorem 6.0.32 For all l < a, ß < oj ( {0 }, ({(0 ,0 )}, { (0 ,0 )} ) ) is the only frame which splits 55(a) ® S5(ß).

Proo f. Clearly ( {0 }, ({(0 ,0 )}, { (0 ,0 )} ) ) splits any 55(a) ® 55(/J). For the other direction we first show

Claim 1. Let Q £ Pr/(55(a)®55(/?)) split 55 (a)®55(0 ). Then Q £ Fr} {55(2)®55(2)).

Proof. If follows immediately from the completeness proof in Kracht & WoLTER [91] that we have for n £ oj:

55(a) ® 55(0) = f } { (S 5 (a ) ® 55 (0 ))(O 2mCI<m)(P r1(A/t2) A Pr2(A lt2))\m > n })

113

Now let Q £ F r j (S 5(2) ® 55(2)) be finite. Clearly there is an n £ u> with 6 0 2mCl(m) (P r 1(yl/t2) A Pr2(Alt2) ) for all m > n.

Hence Th(Q) is not prime in £ (5 5 (a )® 55(/?)).

It follows that the theorem is proved if ({0 }, ( { (0 ,0 )}, {(0 ,0 )} ) ) is the only frame which splits 55(2)® 55(2). It follows from Theorem 6.0.31 that a frame which splits 55(2)® 55(2) is 55-cycle free. The finite and rooted 55-cycle free frames above 55(2) ® 55(2) are {p(m,0)|m € u>} U {<7(m,i)|m > 0,t = 1,2}, where

g(m, 0) := ( 2 m + l , ( « i fl (2m + l ) 2, < 2 n (2m + l ) 2)) for in € w, g(m, 1) := (2m ,$i,s2) with $ 1 := < 1 0 (2m)‘2 and s2 := < 2 H (2m)2, g(m,2) := (2m ,r!,r2) with ^ := <J2 n (2m)2 and r2 := < 1 0 (2m)2, where

< 1 := {(7*, 7i)|7i € tt?} U { (27*, 211 + 1)|» € <*>},< 2 := {(n,n)|n € u>} U {(2n + l ,2n + 2)|n € a;}.

Clearly we have for all infinite sets R C u and i = 0,1,2 that 55(2) ® 55(2) = f|{Th(g(m, t))|m € R}.

Now <7(71, 1) £ F r (y (7n ,0)) for i = 1,2 and n > 0,m € a; because <7( 7*, 1) contains no point x with {y|a\s2y} = { x} and g(ny2) contains no point a: with {y\xriy} = { x } . It follows that the theories I7 i((p (m ,*))) are not prime in £(55(2) ® 55(2)) for m > 0 and i = 1,2. Now consider a frame <7(771,0) with m > 0. It is readily proved that <7(771, 0) £ Fr(p(7i,0)) if 2m - f l / 2ti -f 1. We have

55 (2 )® 55(2) = f){Th(g(n,0))\2m+ 1 /2ti + 1}.

Hence Th(g(m, 0)) is not prime in £(55(2) ® 55(2)). H

7 i?-persistent Subfram e Logics without the F M P

So far we do not have an example of a complete subframe logic without the fmp. Here we give an example of a monomodal complete and elementary subframe logic without the fmp. In Wolter [93] is an example of a finitely axiomatizable bimodal logic with these properties.We list some universal conditions with corresponding modal axioms.Define Q?[qi, .. := < j < n,*i = blank ,7Tj = for j ^ i). We willsometimes abbreviate Q*[q\, . . . , qn] by QJ1 or Q?[q\- Define ptn[qu .. .,?n] := A (O Q f V Q?|l < * < «)•

1. Universal condition E (B \ ) : V s ,y : i < l y = > y < i .Modal axiom: B\.

114 7 R-PERS1STENT SUBFRAME LOGICS WITHOUT THE FMP

2. Universal condition:For all m € w : E (A lm) :=Vy,z : z £ T rm(y), y<fiy,z<fi z => y = z.Modal axioms: A\m := Om(Dp A ->p) A (Dg A ->9) —► (Dp A ->p).

3. Universal condition: E(C 1) := Vx,p, 2,21,22 :2 5 2i , 2i 5 22, 2 ^ 22, x <3 y,y-fly,x < 2,2 < 21,2 <] 22 => X < 21 or X <J 22.Modal axiom Cl := <0(Dp A ->p) A O (Q$ A OCJj A OQ3) —*• (OQ2 v ^Q|)-

4. Universal condition:For all m € u>: £ (A2m) := Vx,y: ITr^x)! > 5,y £ T rm(x), JTr1(y)| > 5 => x < y.Modal axioms: A2m := (Dp Apts[91,.. - ,9s]) Om(pfs[pi,• • -Ps] —1" p)-

5. Universal condition: E(C2) := Vx : ITr^x)! > 4 => x <] x.Modal Axiom: C2 := pt4[9i ,. • •, 94] —*► ( aP~* p)-

6. Universal condition: E(C3) := Vxj,X2,13,14 :ITr^xi)! > 5 ,X! < x2 <J x3 < x4 , i i ^ x3,x j^ x4 ,x3 # x4 =*> |Tr1(x4)| < 3 .Modal Axiom: C3 := (ClQftfl Apt5[p]) - □ (2)( «| [9] - □ (Q ltö - -P*4[p]))).

Define for i = 1,2 sets A’ := {A tm|m € w } and let A := K(B\;Al ,A2;Cl;C2]CS). It is easy bnt tedious to check that A is an A-persistent subframe logic with .Fr(A) = {h £ Fr\h |= E(D ) for D = A ',i = 1,2, D = B\,D = C l,C 2 ,C 3 }. Hence A is a complete and elementary subframe logic.

Theorem 7.0.33 A does not have the fmp.

Proof. Define

$ := □ ((□ ? -> q) —> ->9i A ->q2 A -193)A pfs[Pl,--.,P5] A 0(09 A -.9)

A °((D 9 ~*q)~* 0(Q?[9] A 0(<5|[9] A 0(Q|[9] A 0(C?f[9l A Oph\pi, . • • ,Ps])))))-

We show that -><t> g A but g \=. -><f> for any g £ Frj(A ). Define a frame (/»,<) by h := {3»|n £ w } U (w x {a } ) and

<J := { 3n|n £ u>}2

U {(3n, (3n, a))|n £ w } U {((3n, a), 3n)|n € w }

U {((n ,a ),(n ,a ))|n £ u - {0 } } U {((n ,a ),(m ,a ))||n - m| = 1}

(0,a) (3, a)

( * , « > = **------- ^ "0 3

It is clear that {h,<j ) |= A. Define a valuation ß by ß(q) := h — { ( 0, a )} andß(q*+1) “ {(3n + t, a)|n £ u>} for t = 0,1,2 and ß(pi) := {3 t } for t = 1 ,..., 5. It followsimmediately that (h, <),/?,0 f= <f>.

115Now suppose that (g,<j ) £ F r (A ) and g,ß,x (= <j>. We may assume that x is a root of g. Then |TV*(x)| > 5 and there is an irreflexive point x j £ g with x < x j and x ^ xj. It follows from B1 and C l that x j is the only irreflexive point in g. Define b := Tr^(x) —{x j } . We have

( 1) b H ß(Q'f) = 0 for i = 1,2,3 ( by definition of <f>).

(2) ß(pt5) C b (by j42„,).

(3) For any z £b: |7Y ](z ) - 6| < 1 (by C l).

(4) |2V](xJ)| < 3 (by C 2).

Now xl £ ß(Q\) and there exist xjj £ ß (Q f),x j, £ ß{Ql),x\ £ ß(Q^),z-i £ ß(pt5) with x <1 i j < Xq <3 ip <1 x| <1 zi. It follows with (4) that x j ^ x] and xj^l xj. By ( 1) we have x ^ z\ and by (2) we have x < z\. Hence, by (C3), the picture is as follows:

_1 _ 2 - 3 _ ixo xo xo xi«--- •-----* f•----------------

X Z ]Now we have z\ £ Oß{Q\ A 0(^2 A A A P^s))))- It follows from (3) that thereare x\ £ /?(Q2)>xi £ ß(Q2),x\ £ ß(Q\) and z2 £. ß(ph) with x| < x\ < x f < x\ < z2. It follows from (4) and C3 and the same arguments as above that the picture is as follows:

12

x Z\ z2Again x <1 z2. It should be clear that we can go on in this way and get sequences of different points x},x f,x f,Z i,i £ u in g. Hence g is infinite.H

116 8 INDEX

8 Index

algebra n -~

finitely presentable ~ 30 refined ~

full - 12 rooted ~

«-generated free A -~ 11 55-cycle free ~

subframe ~ 24 T-cycle free ~

subdirectly irreducible ~ 11 t-cycle free ~

arrow subframe 45 tense ~

strict ~ 45 immediate variant

C’-degree o f incompleteness 36 internal cycle

O’-degree o f fmp 36 internal set

C- logic 21 irreducible

C-variety 21 logic

cluster 86 basic elementary ~

degenerated ~ 86 «-com pact ~

proper ~ 86 C -pretabular ~

simple ~ 86 compact ~

~ valuation 88 complete ~

compactness preserving 20 confinal subframe ~

complex 13 connected ~

« - ~ 13 D-persistent ~

consistent 10 elementary ~

depth 47 normal n-m odal ~

inverse ~ 47 n-transitive ~

descendant 88 R-persistent ~

immediate ~ 87 sequentially complete ~

describable operation 20 sequentially «-com pact '

diagram 30 strictly C -complete ~

downward closed 91 tabular ~

finite embedding property 41 tense ~

finite model property 13 lower cover

formula modal depth

C - ~ 20 model

C-splitting ^ 31 n-reflexive iteration

indecomposible ~ 35 n-tree

dual C- ~ 21 reflexive ~

fusion 14 open filter

frame opremum

«-canonical ~ 12 path

cycle free ~ 12 prime

descriptive ~ 13 projectionKripke ~ 12 downward ~

m-transitive ~ 12 upward ~

121212

11112

1108488

1091227

411381

131324721314101013131336397336101154141411291227

2020

117

recognizable 45sequentially derivable 10 spectrum

C-~ 36C-Fine ~ 36fmp-C-Fine ~ 36

splitting 27union-~ 28iterated ~ 28iterated C'-~ 31~ pair 27

strict C-frnp 36subframe 23

<f>-~ 26subreducible 40T-closed 50tense-extension 84t-p-morphism 84transitive closure 52well partial order 35

118 9 LIST OF SYMBOLS9 List of Symbols

£ „ ,£ " 10£A 10K 10A(T),A(<j>) 10□<mV , 10M A n 11(A ,ß) f= <£> 0,M,/3> |= T > 0 11Th(A ),Th( K ) 11VS,,V/,V/.,,- 11A/F, (D ) 11Fr\(n),FTA(p0, . . . ,p K) 11G+ 12A+,a+,U(A) 12^ a (k ) 12Gfr, Rfr, Fr 12Trg(x ),T r? (x ) 129X 12£/r 13®(A«|t < n) 15a I d , a I d 20<t>c 20

21CA 23© Tc, © |c 23Gb,Ab 24B Q A , G Q H 24Sf{Q),Sf{A) 24

24^5/ 25

Sf Con 26T>/p,a/po 27V/F,V/F1/.../Fn 28A/CB,A/CA 31A s 33n»c(F ), 36FncA(& ),fFnc\(B ) 36Spck(B ) 38Xg, V j 40

SA,SATn 42S fn(£ ) 42F <x G ,F <x G 45(F -► C?) 45(T <o C? «- A*) 45dpg(x),dpg(x) 47

50Va< 50^ 51N n,B n ,T r - ,I - 51(<l,|t < n)* 52(T )r 52( ' l l R e f 53(T)n 55T R “ 56Ha,A0 59Pri(G), Pri(A) 71C» 72A+.t,A~.t 84H F ‘ 84<J + , < - , 0 + ,0 - ,G + ,D - 84afW/17] 86c/ (a ),c/u (a ) 86Im g(U ) 870 [{V ,,V 2}/17] 87FD(G),F5(£), £ ,(£ ),£ ;(£ ) 88

11910 List of Logics

Kn 10<8>{A,|i < n) 15T, 55 16K.Altn 19®mK.Altn 19Ver, Triv 26K4, 54, Kb, K.B ] , G, Grz 26K.In 26A .t, A.UU A.U2, A .Id 26K.Trn 26K.Bn 51G. 3 5954.2 6954.1 70A+.t, A- .t 84A+ 84

120 11 REFERENCES

11 References

[88] Bellisima, F.: On the Lattice of Extensions of the Modal Logic K.Altn, Arch. Math. Logic 27: 107- 114, 1988

[83] van Benthem, J. F. A. K.: Modal and Classical Logic, Bibliopolis, 1983

[93] Berarducci, A. & Verbrugge, R.: On the provability logic of bounded arithmetic, Annals of Pure and Applied Logic 61: 75 - 95, 1993

[76] Blok, W.: Varieties of interior algberas, dissertation, University of Amsterdam, 1976

[78] Blok, W.: On the degree of incompleteness in modal logic and the covering relation in the lattice of modal logics, Report 78-07, Dept, of Math., University of Amsterdam, 1978

[80a] Blok, W.: Pretabular Varieties of Modal Algebras, Studia Logica 39: 101 - 124, 1980

[80b] Blok, W.: The Lattice of Modal Logics. An algebraic Investigation, JSL 45: 221 - 236, 1980

[66] Bull, A.: That all normal extensions of 54.3 have the finite model property, ZML 12: 341 - 343, 1966

[77] Esakia, L. & Meskhi, V.: Five critical systems, Theoria 43: 52 - 60, 1977

[74a] Fine, K.: An ascending chain of S4-logics, Theoria 40: 110 - 116, 1974

[74b] Fine, K.: An incomplete logic containing 54, Theoria 40: 23 - 29, 1974

[75a] Fine, K.: Normal Forms in Modal Logic, NDJFL 16: 31 - 42, 1975

[75b] Fine, K.: Some connections between elementary and modal logic, in: Kanger [75], 15-51

[85] Fine, K: Logics containing K4, Part II, JSL 50: 619 - 651, 1985

[76] Gerson, M.: A Neighbourhood frame for T with no equivalent relational frame, Zeitschrift für math. Logik und Grundlagen der Math. 22: 29 - 34, 1976

[89] Goldblatt, R. I.: Varieties of Complex Algebras, Annals of Pure and Applied Logic 38: 173-241, 1989

[92] Goranko, V. & Passy, S-: Using the Universal Modality: Gains and Questions, Journal of Logic and Computation 2: 5 - 30, 1992

[71] Henkin, Monk & Tarski: Cylindric Algebras, Part I, North Holland, Amsterdam, 1971

[75] Kanger, S.: Proceedings of the Third Scandinavian Logic Symposium, North-HoUand, Amsterdam, 1975

121

[90a] Kracht, M.: An Almost General Splitting Theorem for Modal Logic, Studia Logica 49: 455 - 470, 1990

[90b] Kracht, M.: Internal Definability and Completeness in Modal Logic, dissertation, FU Berlin, 1990

[92] Kracht, M.: Even more about the lattice of tense logics, Arch Math. Logic 31: 243- 257, 1992

[91a] Kracht, M.: How Completeness and Correspondence got married, in de Rijke, M. (editor): Colloqium on Modal Logic, Dutch Network for Language, Logic and Information, 1991

[93] Kracht, M.: Splittings and the finite model property, JSL 58: 139 - 157, 1993

[91] Kracht, M. & Wolter, F.: Properties of independently axiomatizable bimodal logics, JSL 56: 469 - 485, 1991

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[74] Maksimova, L. L. & Rybakow, V.: Lattice of normal modal logics, Algebra i Logica 13: 188 - 216, 1974

[75a] Maksimova, L. L.: Pretabular extensions of Lewis 54, Algebra i Logica 14: 28 - 55, 1975

[75b] Maksimova, L. L.: Modal Logics of finite layers, Algebra i Logica 14: 304 - 319,1975

[72] McKenzie, R.: Equational bases and non-modular Lattice Varieties, Trans. Am. Math. Soc. 174: 1 - 43, 1972

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[77] Rautenberg, W .: Der Verband der normalen verzweigten Modallogiken, Math. Z. 156: 123 - 140, 1977

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- 522, 1986

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[93] Wolter, F.: Solution to a problem of Goranko and Passy, to appear in Journal of Logic and Computation

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123

12 German Summary

Bezeichne N n den Verband der normalen n-modalen Logiken. Ein vollständiger Subverband D von Afn heißt Kompaktheit erhaltender Subverband von falls für jede endlich axiomatisierbare Logik A die Projektion nach oben A ] d = p|{A; € £|A' 2 A } wieder endlich axiom at isierbar ist. Die vorliegende Arbeit beschäftigt sich mit Kompaktheit erhaltenden Subverbänden von Afn. Zur modelltheoretischen Charakterisierung Kompaktheit erhaltender Subverbände wird der Begriff der beschreibbaren Operation auf normalen modalen Algebren eingeführt. Es wird gezeigt, daß eine Teilmenge D von Mn genau dann ein Kompaktheit erhaltender Subverband ist, wenn es eine beschreibbare Operation C gibt, so daß genau die zu den Logiken in D korrespondierenden Varietäten abgeschlossen unter C sind. Beschreibbare Operationen werden außerdem benutzt, um Splittings in Kompaktheit erhaltenden Subverbänden von Afu zu charakterisieren.

Wichtigstes Beispiel einer beschreibbaren Operation ist die Operation, die jeder modalen Algebra ihre Subframe Algebren zuordnet. Sei A eine modale Algebra. Für 6 € A läßt sich die Relativierung des booleschen Reduktes von A nach b in natürlicher Weise zu einer modalen Algebra expandieren, die wir als Subframe Algebra von A bezeichnen. Eine Logik heißt Subframe Logik, falls die korrespondierende Varietät abgechlossen unter Subframe Algebren ist. SAfn bezeichne den Verband der n-modalen Subframe Logiken. Die Verbände SATn werden in dieser Arbeit detailliert untersucht.

Für zahlreiche Subverbände von SAfn werden die Splittings bestimmt. Desweiteren wird der Begriff des Fine-Spektrums in natürlicher Weise auf den Verband der Subframe Logiken erweitert. Wir sagen, daß eine n-modale Subframe Logik strikt vollständig ist, falls das Fine-Spektrum dieser Logik in SATn die Kardinalität 1 besitzt. Dieser Begriff dient als Werkzeug zur Lokalisierung vollständiger Subframe Logiken in SATn. Zur Bestimmung der Fine-Spektra von Subframe Logiken werden u.a. die Splittings dieser Verbände genutzt. Für sämtliche monomodalen Standardsysteme sowie für zahlreiche polymodale Logiken wird bestimmt, ob diese strikt vollständig im Verband der Subframe Logiken sind.

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