suszko operator

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The Suszko Operator. Part I Author(s): Janusz Czelakowski Source: Studia Logica: An International Journal for Symbolic Logic, Vol. 74, No. 1/2, Abstract Algebraic Logic: Part 2 (Jun. - Jul., 2003), pp. 181-231 Published by: Springer Stable URL: http://www.jstor.org/stable/20016522 . Accessed: 26/02/2014 08:04 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Springer is collaborating with JSTOR to digitize, preserve and extend access to Studia Logica: An International Journal for Symbolic Logic. http://www.jstor.org This content downloaded from 193.0.101.220 on Wed, 26 Feb 2014 08:04:58 AM All use subject to JSTOR Terms and Conditions

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Page 1: Suszko Operator

The Suszko Operator. Part IAuthor(s): Janusz CzelakowskiSource: Studia Logica: An International Journal for Symbolic Logic, Vol. 74, No. 1/2, AbstractAlgebraic Logic: Part 2 (Jun. - Jul., 2003), pp. 181-231Published by: SpringerStable URL: http://www.jstor.org/stable/20016522 .

Accessed: 26/02/2014 08:04

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Springer is collaborating with JSTOR to digitize, preserve and extend access to Studia Logica: AnInternational Journal for Symbolic Logic.

http://www.jstor.org

This content downloaded from 193.0.101.220 on Wed, 26 Feb 2014 08:04:58 AMAll use subject to JSTOR Terms and Conditions

Page 2: Suszko Operator

JANUSZ CZELAKOWSKI The Suszko Operator. Part I.

Abstract. The paper is conceived as a first study on the Suszko operator. The purpose of this paper is to indicate the existence of close relations holding between the properties of

the Suszko operator and the structural properties of the model class for various sentential

logics. The emphasis is put on generality both of the results and methods of tackling the

problems that arise in the theory of this operator. The attempt is made here to develop the theory for non-protoalgebraic logics.

Keywords: Leibniz congruence, Suszko congruence, sentential logic, protoalgebraic logic, matrix, Suszko-reduced matrix, deductive homomorphism, natural extension.

Introduction

The paper is an attempt of building a general framework for the algebraic study of non-protoalgebraic sentential logics. This framework, which paral lels the well-known hierarchy of protoalgebraic logics based on the Leibniz operator, is centered here on the notions of a logical matrix and of the great est Suszko congruence of a matrix. The mapping Z which assigns to each

matrix its largest Suszko congruence is called the Suszko operator, see ? 1. Equivalently, the Suszko operator for a logic C is the largest of all opera tors for C that is monotone and compatible with the theories of C. This operator plays a key role in this approach. Although the Suszko operator is (set-theoretically) definable in terms of the Leibniz operator (since the greatest Suszko congruence SCM of a matrix M equals the intersection of the Leibniz congruences of the deductive filters of M), the emphasis is put here on the specificity of the Suszko operator and its properties.

In the paper an outline of the theory of the Suszko operator is presented. The results obtained for the Suszko operator are compared with analogous results for the Leibniz operator (see [1]) pointing out those that are natural generalizations of results for the Leibniz congruence that apply to protoalge braic logics. Furthermore, the relationship between the theory of the Suszko operator and the approach to abstract algebraic logic initiated by Brown and Suszko [4] and then developed by Font and Jansana [15] is underlined.

Special Issue on Abstract Algebraic Logic - Part II Edited by Josep Maria Font, Ramon Jansana, and Don Pigozzi

Studia Logica 74: 181-231, 2003. ? 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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182 J. Czelakowski

Let S be a fixed sentential language and let T be a set of sentential formulas in S (i.e., T is a theory in this language). QT is a binary relation on S defined by:

(a, /) C QT iff, for every sentential formula (o E S and every

variable x occurring in p, p(x/o) E T iff (x/l) C T.

The relation QT is known to be the largest congruence of S compatible with T ([20]). After Blok and Pigozzi [1], the congruence QT is called the Leibniz congruence on S over T. The mapping Q which to each theory T assigns the congruence QT is called the Leibniz operator (on S).

The definition of the Leibniz congruence QT is, in fact, a linguistic at tribute of the theory T, reflecting the grammatical structure of the language. (QT may be regarded as the synonymy relation between the expressions of S relative to the theory T.) The fact that QT is an absolute notion, depend ing only on the structure of the language, and not on the logic admitted in the language, has various consequences. (In this paper the notion of a logic is identified with the notion of a structural consequence operation C in a sentential language S.) The definition of a protoalgebraic logic (Definition 0.8) establishes the relationship between the grammatical structure of some theories in S (viz. the theories of C) and the logical structure codified in this language, i.e., the totality of all rules of inference admitted in S. Strictly speaking, protoalgebraicity means that any two sentences synonymous with respect to a closed theory T are logically equivalent modulo T. Equivalently,

C is protoalgebraic if Q is monotone on the theories of C. These links are strong enough that, as shown in [8] every protoalgebraic consequence oper ation C is reducible, through a suitable deduction theorem, to the logical system C(0). Thus one may say that protoalgebraic logics are determined by their systems of logical theses. This "determination"is not uniform, how ever - two different protoalgebraic logics may have the same sets of theses. For example, each normal modal system determines in a natural way two

distinct modal protoalgebraic consequence operations: the normal and the

quasi-normal modal logic. There is an array of logics, possessing interesting metalogical properties,

in which the relations holding between the consequence operation and logical theses are less tight. (In particular, the logics with the empty sets of theses

fall into this category.) These logics are mainly non-protoalgebraic. Here are examples:

1. Some logics determined by two-element matrices M. (We admit that

only one element is designated in M.)

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The Suszko Operator. Part I. 183

A result established by Rautenberg [23] says that every logic determined

by a two-element truth-table is finitely based. This theorem cannot be proved by means of the methods worked out by the theory of protoalge braic logics because the logics examined by Rautenberg are mainly non protoalgebraic. For example, a logic C complete with respect to a two element matrix is protoalgebraic iff it is an extension of either the purely equivalential or the purely implicational fragment of classical consequence (see e.g. [7, Exercise 1.2.1]). Shortly speaking - there are only few protoal gebraic logics among the two-valued logics.

2. In particular, the conjunctive-disjunctive fragment K(AV) of classical

logic, i.e., the restriction of the consequence operation of classical logic to the language which involves only the connectives A, V, is not protoalgebraic. The latter logic, which obviously has no tautologies, was extensively investigated by many logicians, e.g. by Dyrda and Prucnal [11] and the logicians from Barcelona (Font and Verdui [13]).

K(AV) is complete with respect to the two-element (distributive) lattice in which the unit element is designated. One may expect that the class

Mod*(K(A,v)) of its Leibniz reduced models (or, more precisely - the class of algebra reducts of the members of Mod*(K(Av))) coincides, up to isomor phisms, with the class of distributive lattices. This supposition was dashed by Font, Guzmain and Verdu' [12]. They have shown that the reduced models of K(Av) are identical with distributive lattices A with unit 1 in which, for all distinct a,b c A with a < b, there is a c C A such that aVc #4 1 and

b V c = 1. Thus, for example, the three-element chain is the algebra reduct of no reduced model for K(A,v).

3. The logic J(A v-), the implicationless fragment of intuitionistic logic, is also known to be non-protoalgebraic ([2], [25])

4. A logic (S, C) is selfextensional iff the Suszko congruence SC(0) is characterized in the following way:

a =3 (EC(0)) iff C (a) = C0().

The above notion was introduced and extensively studied by Wojcicki [31], [32] (see also Czelakowski [7]). The characteristic feature of selfexten sional logics is that they are determined by the so-called referential matrices.

We shall discuss the latter notion in outline.

Suppose T is a non-empty set. The elements of T are called points of reference (whatever they can be: time instances, states of affairs, possible

worlds etc.) A referential algebra is any algebra similar to the language S

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184 J. Czelakowski

whose underlying set is a subset of {O, 1}T, i.e., the members of the algebra are functions from T into {O, 1}.

Given a referential algebra W on T, for every point t C T we define:

Dt := {f C W: f(t) = 1};

D := {Dt : t C T}.

The pair (W, D) is then called a referential (or: ramified) matrix for S. The logic determined by (W, D) is meant to be the consequence C = inf{Ct : t C T}, where Ct is the consequence operation determined by the usual matrix (W, Dt).

Wojcicki's Theorem says that C is selfextensional iff C is determined by a referential matrix.

The concept of referential matrix is of great theoretical importance due to its relationship to other forms of referential semantics, e.g., the relational or neighbourhood semantics. This issue is discussed in [32].

We mention here that there are numerous examples of selfextensional logics that fail to be protoalgebraic. The logics K(Av) and J(AV,-) are among them.

5. W jcicki's consequence operations corresponding to the well-known entailment and relevance systems as E or R are not protoalgebraic. Let L be either the system E or R. Let CL be the mapping defined on the power set of the language of L by means of the stipulation:

a C CL(X) iff X is non-empty and for some finite set {1, ... , an} C XI

cl A ...A A a\n - a C L;

CL(0) = 0.

CL is a well-defined finitary and structural consequence operation (see e.g. [32]). The logics CL are not protoalgebraic however. In the case of R, the logic CR has been studied, under the name WR, by Font and Rodriguez [17]. They determined the reduced matrices for CR and the algebra reducts

of reduced CR-models.

In the literature the notion of the Leibniz operator has been much dis cussed in connection with the theory of protoalgebraic logics ([1, 2, 3]). However, to the best of our knowledge, no adequate applications of this op erator have been given for non-protoalgebraic logics at a level of generality comparable to the theory developed by Blok, Pigozzi and others.

The present paper is the first step towards establishing a certain general classifying scheme of deductive systems, parallel to that characteristic for

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The Suszko Operator. Part I. 185

protoalgebraic systems (see e.g. [7] for details). This scheme should satisfy the following two postulates:

(a) the scheme comprehends all sentential logics,

(b) when relativised to the class of protoalgebraic logics it yields the well known protoalgebraic logics hierarchy.

The key role will be played here by the notion of the Suszko operator E. The theory of the Suszko operator will enable one to classify, in a natural way, all deductive systems according to a certain general scheme - through referring to some self-imposing properties of the operator E such as continuity or

infectivity. (We mention that the Suszko operator is always monotone.) This classification procedure will be parallel to that described in [3] and in [7] for protoalgebraic systems. The only difference is that the Leibniz operator will be replaced by the Suszko one. It should be mentioned that the choice of the Suszko operator as a basis of classifying scheme is not haphazard - in case of protoalgebraic systems the two operators coincide (Theorem 1.10).

0. Background

Sentential languages, logics and rules of inference

A standard sentential language S is the absolutely free algebra freely gen erated by a countably infinite set Var(S) = {x : n E N} of free gener ators (variables) and endowed with a countable set of finitary operations (connectives). (N is the set of natural numbers.) We note that each stan dard language is countably infinite. (In some metalogical applications also nonstandard sentential languages are considered. These are absolutely free algebras generated by uncountable sets Var(S) of variables.). The members of the language are called sentential formulas, or simply sentences, and the language itself is also called the formula algebra. To simplify notation we adopt the convention that algebras and their universes are denoted by the same symbol. In the context of this paper such an ambiguity in the meaning of symbols should not cause difficulties. Accordingly, each sentential lan guage and the set of its sentences are denoted by the same symbol S (with indices if necessary).

A logic is a pair

(Si C),

where S is an arbitrary standard sentential language and C is a structural consequence operation on S ([30], [21]). This means that for all X, Y C S

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186 J. Czelakowski

and all endomorphisms (substitutions) e: S -* S, the following conditions hold:

(a) X C C(X) (reflexivity)

(b) X C Y implies C(X) C C(Y) (monotonicity)

(c) C(C(X)) C C(X) (idempotency)

(d) eC(X) C C(eX) (structurality).

The cardinality of a logic (S, 0) is the least infinite cardinal number ,i such that C(X) = U{C(Y) : Y C X and JYJ < j,4 for all X C S.

The cardinality of (S, 0) is either No or N1. A logic (S, C) is finitary if its cardinality is equal to No.

If X is a finite set of sentential formulas and 9p, . . ., pn is a fixed enu

meration of the elements of X, then we often write C(ypj,..., <Pn) instead of

C(X). We also use C(X, (p) as an abbreviation for C(X U {y4). As is customary, logics are often identified with their consequence oper

ations. Furthermore, the terms "logic "and "deductive system "are treated here as synonymous.

Th (C) is the family of all theories of C. Thus T C Th (C) iff C(T) = T.

Given a pair (X, a), where X C S and a C S, we define

X/a := {(eX, ea) : e is a substitution in S}.

The set X/a is called the sequential rule of inference in S determined by (X, a). The pair is called a scheme of X/a. If X is finite, the rule X/a is called finitary. In the case when X is empty, the rule X/a is called axiomatic.

Let Q be a set of sequential rules. A set T C S is closed with respect to the rules of Q (T is Q-closed, for short), if eX C T implies ea E T for all rules X/a in Q and all substitutions e in S. The family of all subsets of

S which are closed with respect to the rules of Q is nonempty and forms a

closure system. CQ stands for the consequence operation on S corresponding to this closure system. Thus

CQ(X) := {T: T is Q-closed and X C T}.

Every structural consequence operation C on S can be represented as

CQ for some set Q of sequential rules. (Take Q to be the collection of all

rules X/a such that a C C(X).) If the rules in Q are finitary, the consequence CQ is finitary as well. In

this case the consequence CQ is conveniently characterized in terms of Q proofs. Let T C S and p E S. A Q-proof of cp from T is a finite sequence of

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The Suszko Operator. Part I. 187

sentential formulas of S

'Po *** (Pn

of which each is either a member of T or is obtained from previous ones by an application of a rule of Q and o is the last element of the sequence. I.e.,

,On is identical with p and for every i, 0 < i < n, either (pi T or there

exists a set J C {O. ... , i - 1}, a rule X/a and a substitution e in S such that

f{j :j E J} = eX and poi is identical with ea. We then have: so E CQ(T)

iff there exists a Q-proof of p from T.

If C is a finitary logic, then any set Q of finitary sequential rules which defines C, i.e., C = CQ holds, is called an inferential base of C.

Matrices and deductive filters

A logical matrix (a matrix, for short) is any pair M = (A, D), where A is an algebra and D is a subset of A. The set D is called the truth-predicate of M or the set of designated values of the matrix M. The algebra A is referred to as the algebra reduct of M.

Let S be an arbitrary but fixed standard sentential language. Any al gebra similar to S is called an S-algebra, for brevity. A matrix M is called a matrix for S if the algebra reduct A of M is an S-algebra. Homomor phisms from S to the algebra A are customarily called valuations (of S) in the matrix M.

Each class K of matrices for S defines the structural consequence oper ation KW= in S, where

a E K`(X) iff, for every matrix M = (A, D) in K and for every valuation h: S -- A, h(X) C D implies h(a) E Di

for all X C S and a E S. Thus a E K'(X) iff, for every matrix M of K, each valuation h in M which validates the sentences of X in M also validates a} in M.

If K consists of a single matrix, K = {M}, the consequence KW is usually denoted by Mt.

Let (S, C) be a logic. A class K of matrices for S is called adequate for C if KW = C. In this case we also say that C is determined by the class K or that C is complete with respect to K.

A matrix M = (A, D) is called a model for C if it has the property that for all X C S and a C S if a E C(X) then for every valuation h: S -* A,

h(X) C D implies h(a) c D, i.e., h validates a in M whenever it validates X in M. Evidently, if a class K is adequate for C, then each matrix in K is

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188 J. Czelakowski

a model for C. Mod(C)

denotes the class of all models for C.

L(C)

stands for the set of matrices of the form (S, T), where T C Th (C). Each such a matrix is a model for C. Hence L(C) C Mod(C). L(C) is called the Lindenbaum family (or the Lindenbaum bundle) for C. The members of

L(C) are called Lindenbaum matrices for C. Let A be an S-algebra. A set D C A is called a deductive filter of the

logic C in the algebra A (D is a C-filter, for short) if the matrix (A, D) is a model for C. The family

FiC(A)

of all C-filters in A is non-empty since it contains A, the universe of the algebra. The C-filters in A form a closure system. Consequently, Fic(A) is a complete lattice with inclusion as the lattice order. For X C A,

CA(X)

denotes the least C-filter in A that includes the set X. CA is thus a closure

operator on the algebra A. In particular, if A is the language S of C, Fic(A) coincides with Th (C)

and CA = C.

If C is finitary, then the lattice Fic(A) is algebraic for all S-algebras A and the compact elements of Fic(A) are exactly the finitely generated C-filters in A, i.e., the filters of the form CA(X) with X finite. These observations are easy consequences of the following facts.

Let Q be a set of sequential rules in S and let A be an S-algebra. A set

D C A is said to be closed with respect to the rules of Q if h(X) C D implies

h(a) c D for all homomorphisms h: S -> A and all rules X/a in Q. It is

easy to verify that the family of all Q-closed subsets of A forms a closure

system. Furthermore, this system is finitary if the rules in Q are finitary.

THEOREM 0.1. Let (S, C) be a logic and let Q be a set of sequential rules

such that C = CQ. Then, for any algebra A and any set D C A, D is a

C-filter in A if D is closed with respect to the rules of Q. Consequently, the closure system Fic(A) coincides with the family of all Q-closed subsets

of A.

We have the following description of the filter CA(X):

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The Suszko Operator. Part I. 189

PROPOSITION 0.2. (A). Let (S, C) be a sentential calculus and let Q be a set of sequential rules such that C = CQ. Then, for any S-algebra A and any set X C A, CA(X) is the smallest subset D C A which includes X and is closed with respect to the rules of Q.

(B). Furthermore if (S, C) is finitary and the rules of Q are finitary, then the following equivalence holds: b E CA(X) iff there exists a natural number

n > 1 and a sequence

(1) (Cl C2 **.. CO)

of elements of A with Cn = b such that for every i, 1 < i < n, either cj E X

or there exists a set of indices Ji C {1, ... ,i - 1}, a rule Y/a in Q and a homomorphism h: S -> A such that

(2) {hey: y c Y} = {cj: j E J} and ha = ci.

The easy proof is omitted. i

The sequence (1) is called a Q-proof of b from X in A. If (2) holds, we say that ci directly follows from {cj: j E Ji} by means of Y/ca.

A sequence (ci, c2,. . ., iC) is called a C-proof of b from X in A if it is a

Q-proof for some set Q of rules of C.

Congruences and homomorphisms

Let M = (A, D) and N = (B, E) be matrices for S. Hom (M, N) is the set of all homomorphisms from M to N. Thus h C Hom (M, N) if h is a homomorphism from the algebra A to B and h(D) C E. If h: M -> N is a homomorphism, then the set ker(h) := {(a, b) E A2: h(a) = h(b)} is called the relational kernel of h. ker(h) is a congruence of the algebra

A of M. A homomorphism h C Hom (M, N) is called strict if furthermore h(A - D) C B - E. Thus, an algebraic homomorphism h: A -* B is strict

iff h-1 (E) = D. Let M = (A, D) be a matrix for S. For each congruence 4F of A, the

quotient matrix M/1? is defined as the pair (A/J@, D/1?), where A/4? is the quotient algebra and D/14 := {[a]: a C D}. ([a] is the equivalence class of a relative to (.) The mapping k.: A -+ A/? defined by: kb(a) := [a], for all a c A, is a surjective homomorphism from M to M/4?. kD is called the natural (or canonical) homomorphism from M to M/I. Evidently,

ker(kb) = C A congruence '1 of the algebra A is compatible with the set D if a c D

and (a, b) C P imply b C D for all a, b E A. Thus (D is compatible with D iff

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190 J. Czelakowski

D is the union of some equivalence classes of (1. The congruences compatible

with D are also called strict congruences of the matrix M.

PROPOSITION 0.3. Let M = (A, D) and N = (B, E) be matrices for S.

(i) Let h E Hom (M, N). If h is strict, then the relation ker (h) is a

strict congruence of M.

(ii) If h C Hom (M, N), h(D) = E and ker(h) is a strict congruence of

M, then h is strict.

(iii) If 4) is a congruence of the algebra reduct of M, then k' is a strict

homomorphism from M to M/4? if ( is a strict congruence of M.

(iv) If h E Hom (M, N) is surjective and strict, then the quotient matrix

M/ker (h) is isomorphic with N.

Let (S, C) be a logic and let M = (A, D) be a matrix for S. We define

Fic(M) := {F E Fic(A): D C F}.

Fic(M) is the set of C-filters of M. For any subset X of A we define:

CM(X) := CA(D U X);

this filter is called the C-filter of M generated by X. If X = {a}, we write

CM(a) instead of CM({a}); such a C-filter of M is called principal. The following proposition is due to Blok and Pigozzi [1]:

PROPOSITION 0.4. Let (S, C) be a logic. Let M = (A, D), N = (B, E) be

matrices for S and let h : M -* N be a surjective homomorphism. Let

F E FiC(M) and G, GI, G2 E FiC(N). Then:

(i) CN(h(F)) E Fic(N) and h-1(G) E Fic(M);

(ii) CN(h(h-1(G))) = h(h-1(G)) = G;

(iii) h-1(CN(h(F))) = h-1(h(F)) = F if h-(E) C F and ker(h) is

compatible with F;

(iv) h-1(GI n G2) = h-1(Gi) n h-1(G2);

(v) CN(h(CM(X))) = CN(h(X)) for any subset X of A.

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The Suszko Operator. Part I. 191

The Leibniz operator

DEFINITION 0.5. Let A be an algebra and let D be a subset of A. The Leibniz congruence on A for D, denoted by QAD, is the greatest congruence on A that is compatible with D.

THEOREM 0.6. For every S-algebra A and every set D C A, the Leibniz

congruence QAD exists. Furthermore there holds the equivalence:

(a, b) C QAD if, for every sentential formula ~o(x, ul,... ,u n) and every sequence ei, ... , en of elements of A, cp(a, el,... , en) C D iff

(p(b, el, i., en) C D.

(Here o(a, e1, ... , en) is the value of (p in A when the variables x, ul, .. ,n are assigned the elements a, el,... ,en, respectively.)

For the proof, see e.g. [1] or [5], [7]. U

Let M = (A, D) be a matrix. When the truth predicate of M is not

made explicit, the congruence QAD is denoted by QM. The congruence QM is also called the Leibniz congruence of M.

A matrix M is Leibniz-reduced, or simply, reduced, if QM is the identity relation.

For the formula algebra S we have the following description of QST due

to Los [20]. (Following the common practice we usually write QT dropping the subscript when S is clear from context.)

PROPOSITION 0.7. For each set T C S:

(a, 3) c QT if, for every sentential formula o G S and every variable x occurring in so, ,o(x/a) c T iff o(x/lo) c T.

(p(x/a) is the sentence obtained from p by uniformly substituting the vari able x by the sentence a.)

For the proof, see e.g. [7].

As a particular instance of the general definition, the congruence QT is called the Leibniz congruence on S for T.

Let M = (A, D) be a matrix. The Leibniz reduction of M, or simply, the reduction of M, is the quotient matrix M/QAD :_ (A/QAD, DIQAD).

This matrix is also denoted by M/QM. The canonical homomorphism from M to M/QM is strict and QM is the greatest strict congruence of M.

Trivially, if M is Leibniz-reduced, then it is isomorphic with M/QM because QM is the identity relation. Hence every Leibniz-reduced matrix is isomorphic with its Leibniz-reduction. Conversely, it is not difficult to prove that for any matrix M, the quotient matrix M/QM is Leibniz-reduced.

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192 J. Czelakowski

For any class K of matrices for S, we define

K* :- {M/QM: M E K},

the class of the Leibniz-reductions of the matrices of K. In particular,

Mod* (C)

is the class of the Leibniz-reductions of models of C. In turn,

L*(C)

is the class of the Leibniz-reductions of Lindenbaum models for C. Letting I(K) to denote the class of all isomorphic copies of the members

of K, we see that I(Mod*(C)) is the class of all Leibniz-reduced models of C. It is not difficult to prove that I(L*(C)) is the class of all countable reduced models of C.

An operator for a deductive system C is any mapping 0 defined on Th (C) such that OT is a congruence on the formula algebra for each T C Th (C). The function Q whose domain is restricted to Th (C) and which assigns the congruence QT to each theory T e Th (C) is called the Leibniz operator for C.

In the methodology of the deductive systems it is convenient to have the operator Q defined in a much more general context - given a deductive system C, the class of mappings QA with A ranging over arbitrary S-algebras A, and where the domain of each QA is the family FiC (A) of deductive filters on A, is called the extended Leibniz operator for C.

Protoalgebraic logics.

DEFINITION 0.8. A logic (S, C) is protoalgebraic if, for all sets T C S and all sentential formulas a, /3 E S, a /- (mod QC(T)) implies C(T, a) = C(T, 3).

There are known several equivalent characterizations of protoalgebraicity which give a better insight into this notion. We mention here a few of them.

An operator ( for C is monotone (on Th (C)) if for every pair T1, T2 E Th(C), T1 C T2 implies OT1 C 0T2.

A logic (S, C) has the Correspondence Property if, for every strict homo morphism h: M -* N between models of C and every filter F C FiC(M), F = h-1h(F).

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THEOREM 0.9. For any logic (S, C) the following conditions are equivalent:

(i) C is protoalgebraic.

(ii) The Leibniz operator Q is monotone (on the family Th (C)).

(iii) For any S-algebra A and any filter D E Fic(A), a -b (mod QAD) implies CA (D, a) = CA(D, b), for all a, b E A.

(iv) The extended Leibniz operator is monotone, i.e., QA is monotone on the family Fic(A), for every algebra A.

(v) (S, C) has the Correspondence Property.

We note here one important consequence of the Correspondence Prop erty.

COROLLARY 0.10. Let (S,C) be a protoalgebraic logic and let h: M - N be a strict surjective homomorphism between matrix models for C. Then the lattices of C-filters of M and N are isomorphic.

PROOF. Indeed, the mapping which to each C-filter F of N assigns the set h-1(F) is the required isomorphism.

1. The Suszko operator

DEFINITION 1.1. Let A be an S-algebra and let C be a family of subsets of A. For each set D C A we define:

ECD:= {QAF: F C C & D C F}.

The congruence ECZD is called the Suszko congruence on A for D over the family C.

We admit that if the set {F : F C C & D C F} is empty, then ECiD =

AxA.~~~~~~~~~~~~~ A x A.

The following observations directly follow from the definition of the Suszko congruence:

PROPOSITION 1.2. Let A be an S-algebra and let C be a family of subsets of A. Then:

(1) SC D C QAD for every set D C C.

(2) If D = n{F : F C C & D C F} then the congruence ECjD is compatible with D.

(3) If D1 C D2, then SD1 C YCjD2, for all D1, D2 C A.

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PROOF. (1) and (3) are immediate. Note that (3) states that EC, treated as a mapping defined on the power set of A, is monotone. As to (2), assume a _ b (mod EAD) for some a,b E A. Hence a b (mod QAF), for all F E C such that D C F. This implies that a E F iff b E F, for all F E C such that D C F. Consequently, a C {F: F C C & D C F} iff bE C{F: F C C & D C F}. As D= f{F: F E C & D C F}, we get:

a E D iff bED. E

We note that unlike the definition of the Leibniz congruence QAD, the definition of ECjD is strictly relativised to the family C. Consequently, the congruence YCD may vary on passing from one family C to another. In fact, if C and C' are families of subsets of A and C C C' then SCZD C EC D for all sets D C A. We also note that the definition of ECZD is algebraic and not metalogical in the sense that it does not refer to any logical system

whatsoever. However, when we take into account various applications of the above notion in metalogic, we restict ourselves to the Suszko congruences over the families of deductive filters Fi C (A) for arbitrary logics C.

If (S, C) is a logic and A is an S-algebra, then for each set D C A the Suszko congruence on A for D over the family Fic(A) is denoted by ECD. Thus

EC:D =flQAF: F E Fic(A) & D C Ff,

for all D C A.

The following theorem characterizes the Suszko congruences over Fic (A):

THEOREM 1.3. Let (S, C) be an arbitrary logic and let A be an algebra similar to S. Then for every set D C A and every pair a, b e A:

(1) a- b (mod ECD) iff for every sentence O(X, Ul,... ,uk) C S and for every string el,... v ek of elements of A, CA(D, fo(a, e C,... , ek)) = CA(DI p(b, el, ... I ek)).

Furthermore the following conditions hold:

(2) YjCD = YZCCA(D) for all D C A;

(3) For any C-filter D C A, YiCD is a congruence on A compatible with D. Consequently, YjCD C QAD.

(4) If D1 C D2, then YZCD, C ECD2, for all D1,D2 C A;

(5) a b (mod YZCD) implies CA(D, a) = CA(D, b), for all a, b C A and all D C A.

[Recall that CA(X) is the smallest C-filter on A that includes the set X.]

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PROOF. (1). Let D C A. We have: EZCD := f{QAF : F E Fic(A) & D

C F}. Hence, for every pair a, b E A:

a _ b (mod ECD) iff

VF c Fic(A) (D C F X a _ b (mod QAF)) iff

(*) VF E Fic(A) (D C F (V(X,Ul, .*,Uk) E SVe = el, ,ek C A

(W (a, e) E F X Wp(b, e) C F))).

Assume ab C A and a b (mod ECD). Fix (x,ul, . . . ,uk) C S and e=el, .. ., ek C A. Let Fa :=CA(D, p(a, e)). As Fa E FiC(A), D C Fa, and

(p(a, e) c Fa, (*) implies that O(b, e) C Fa, i.e., W(b, e) C CA(D, p(a, e)). By

a symmetric argument (for Fb := CA(D, (o(b, e))), we obtain that p(a, e) &

CA(D, W (b, e)). Thus

CA(D, o (a, e)) - CA(D, W (b,e))

for all (x,ul,. . . ,uk) E S and all e= el,... ,ek E A.

Conversely, assume that CA(D, Wp(a, e)) - CA(D, (o(b, e)) for all Wo(x, ui, ... , uk) E S and all e = el, .. ., ek E A. This readily implies that:

if F E Fic(A) and D C F then (Q (a, e) c F W (b, e) E F).

So a _ b (mod QAF) whenever F E Fic(A) and D C F. This means that a _ b (mod ECD).

This completes the proof of (1). (2) follows from the definition of the Suszko congruences YjCD and YjCCA(D) and the fact that CA(D) is the intersection of all C-filters on A that include D. (3) and (4) follow from

Proposition 1.2.(2),(3). In turn, (5) follows from (1) (it suffices to take as (x, u1, . . ., Uk) the variable x). U

If (S, C) is a deductive system, then the family Fic(S) of deductive filters on the formula algebra S coincides with the family Th (C). For each set T C S. the Suszko congruence on S for T over the family Th (C) is denoted by ECT or simply by ZCT when S is clear from context. In the situations when the system C is also clear, the congruence ECT is simply called the Suszko congruence of T. Thus:

EZT := {QsT: T' E Th(C) & T C T'},

for all T C S. Evidently, YCT C QT for every theory T E Th (C).

COROLLARY 1.4. Let (S, ) be an arbitrary logic. Then for every set T C S and every pair a, i EC S:

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196 J. Czelakowski

(1) o /3 (mod ECT) if C(T, p(x/a) = C(T, o(x//3) for every sentence p E S and every variable x C Var (Q).

Furthermore

(2) ZCT = 0CC(T) for all T C S;

(3) For every theory T E Th (C), the congruence YCT is compatible with

T.

(4) If T, C T2, then ZCTi C ECT2, for all TI,T2 C S;

(5) a _ (mod ZCT) implies C(T,a) = C(T,/3), for all a,/3 E S and

all T C S.

PROOF. (1). Use Theorem 1.3.(1) and the fact that for the formula algebra

S and any T C S and a, : E S the conditions

(a) for every sentence p(x,uI i,... , Uk) E S and every string 6i, , k of

elements of S, CA(T, a (a,6i 61, ,6k)) = CA(T, p(, 61,... ,6k)) and

(b) C(T, ~o(x/a) = C(T, y(x//3) for every sentence p C S and every variable x E Var('p)

are equivalent.

(2) - (5) follow from Theorem 1.3. E

If M = (A, D) is a matrix in Mod(C), we often write

SCM:= ECD

and call SCM the Suszko congruence of the matrix M. The function EC (with domain Th (C)) which assigns the congruence

ZCT to each theory T C Th (C) is called the Suszko operator for C. For each S-algebra A we define EC to be the mapping assigning the

congruence ZCD to each C-filter D C A. The class of the mappings EC

(each with the domain Fic(A)) with A ranging over all S-algebras is called

the extended Suszko operator for C. It follows from Theorem 1.3.(4) that for

every deductive system C, the extended Suszko operator for C is monotone,

i.e., for every algebra A, the mapping EC is monotone on Fic (A). The history of the Suszko operator is as old as the history of non-Fregean

logic. The idea of distinguishing between Fregean and non-Fregean logics is mainly due to Suszko [27], [28]. The main feature of non-Fregean logic is the distinction made between semantic reference (the meaning, or the

denotation) of a sentence and its truth-value. In the logical systems de

fined by Suszko, such as e.g. his sentential calculus with identity (SCI), the

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distinction between reference and truth-value is embodied in a new binary connective _ called the identity connective. Connecting two sentences by the identity _ expresses the fact that these two sentences have the same se

mantic correlate while the equivalence connective +-+ expresses the fact that

the two sentences have the same logical value. If S is the language of the classical sentential calculus enriched with

the binary connective _ and C is an arbitrary logic defined in S, then possesses the following meaning in C: for any T C S.

(*) a -B C 0(T) iff C(T, (p(x/a)) = C(T, (p(x//3)) for every p E S and every variable x E Var((p).

According to (*) two sentences a and are identical on the basis of T (i.e., they refer to the same thing) iff a and : are mutually exchangeable in all possible contexts o (represented here by sentential formulas of S) on the basis of T.

One may now entirely abstract from the syntactical structure of the lan guage S, i.e., from the fact whether the language S contains the connective - or not, and treat the definiens of (*) as a condition which defines a binary relation between formulas of S. In other words, for any sentential language S and for any T C S we may define the binary relation YCT on S by the stipulation:

(ca, 3) E ECT iff C(T, W(x/a) = C(T, o(x/3) for every sentence

W E S and every variable x C Var (p).

This is the original definition of YCT given by Suszko [29]. In view of Corollary 1.4. this definition is, of course, equivalent to the one admitted in this paper. Suszko has initiated investigations of the relations ZCT for the theories T of various deductive systems. During the Autumn School of Logic, organized by the Section of Logic of the Polish Academy of Sciences in November 1977, Suszko delivered a series of talks in which he presented a number of observations on ZCT. Some of them are included into the following theorem:

PROPOSITION 1.5. Let (S,C) be an arbitrary logic. Let T C S. Then the following conditions hold:

(6) a - (mod ?CT) implies ea e/ (mod YCC(eT)), for every surjective substitution e: S -- S and all a, / E S.

(7) Zce-1T C e-ZECT, for every surjective substitution e.

(8) If T is finite and a, C c S, then a /3 (mod ?CC(T)) implies that ea e/3 (mod YCC(eT)) for every substitution e: S -? S.

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198 J. Czelakowski

(9) The congruence YCC(0) is fully invariant, i.e., a /3 (mod ZCC(0)) implies that ea e/3 (mod YCC(0)) for every substitution e: S -* S and for all sentences a, 43.

PROOF. To show (6), assume a 3 (mod ?CT) and let e be a surjec tive substitution in S. Then C(T, (p(a, al, . . . I/k)) = C(T, (p(3 o-il ... , k))I for all O(x,yl,. ,Yk) and all strings -y = 7, ... ,Yk of sentences. Hence, by structurality, C(eT, y9(ea, e-y)) = C(eT, ~o(eo, e-l)). As e is surjective, the last equality is equivalent to C(eT, ~o(ea, 6) = C(eT, o(e, 6), for all

bo(x, yl,... , yk) and all strings 6 =k,..., k of sentences. Thus ea _ e/3

(mod ?CC(eT)). (7) directly follows from (6). The proof of (8) is produced by way of a

suitable modification of (6). The details are left to the reader, see also e.g. [7, Theorem 1.5.5].

(9) was originally proved by Suszko. It is a consequence of (8), see also e.g. [32, Lemma 1.7.4] or [7, Theorem 1.5.5 and Remark 1.5.6].

The following result can be easily obtained from Theorem 1.3.(1):

THEOREM 1.6. Let (S, C) be an arbitrary logic and let A be an algebra similar to S. Suppose ( is a mapping defined on FiC(A) to the set Con(A) of congruences of A with the following property:

(*)A a b (mod OD) implies that CA(D, a) = CA(D, b),

for all D C Fic(A) and all a,b C A. Then OD C Y3CD, for every D E

FiC(A)

Thus the above theorem together with Theorem 1.3.(5) state that YjCA is the largest (in the sense of inclusion) of all mappings 0: FiC (A) -- Con (A) satisfying (*)A.

PROOF. Let a, b c A and a b (mod OD). Then (p(a, ei,.. ,ek) p(b,ei,... ,ek) (mod OD) and therefore, by (*)A, CA(D,SO(a,el, ,ek))

= CA(D, (p(b, el, ... , ek)) for every sentence p(x, u,... , k) in S and for ev

ery string el, . . ., ek of elements of A. Hence a _ b (mod ECD) by Theorem

1.3.(1).

COROLLARY 1.7. Let (S, C) be an arbitrary logic. EC is the largest of all

operators 0 for C with the following property:

(*)s for every pair of sentential formulas a, /3 and for every theory T E Th(C), a _ /3 (mod OT) implies C(T, a) = C(T,/3).

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The Suszko Operator. Part I. 199

I.e., EC satisfies (*)s and if C satisfies (*)s, then CT C ZCT for all T E

Th(C). E

A mapping C: Fi C (A) -- Con (A) is monotone if, for all D1, D2 E

Fi c(A), D1 C D2 implies OD1 C OD2. A mapping C : Fi c(A) -- Con (A)

is compatible with the C-filters on A if, for every filter D c Fic(A), the

congruence OD is compatible with D.

The following theorem univocally characterizes the mappings EC:

FiC(A) -- Con(A), for all S-algebras A:

THEOREM 1.8. Let (S, C) be an arbitrary logic and let A be an algebra similar

to S. EC is the largest of all mappings C : FiC(A) -* Con (A) which are

monotone and compatible with the C-filters on A.

PROOF. We first notice that, by Theorem 1.3.(3)-(4), the mapping EC is monotone and compatible with the C-filters on A.

CLAIM. Let C: Fic(A) -- Con (A) be a mapping which is monotone and

compatible with the C-filters on A. Then C satisfies the condition (*)A of

Theorem 1.6. PROOF OF THE CLAIM. Let D E Fic(A) and assume that a -b (mod OD)

for some a,b c A. As D C CA(D,a), we have that OD C CCA(D,a) by

monotonicity. Hence a -b (mod CCA(D, a)). As OCA(D, a) is compatible with CA (D, a), we therefore have: a E CA (D, a) iff b C CA (D, a). So b E

CA(D, a). Since D C CA(D, b), a similar argument gives that a E CA(D, b). Thus CA(D, a) = CA(D, b). So (*)A holds for C. This proves the claim.

The thesis of Theorem 1.8 directly follows from Theorem 1.6 and the above claim.

For the formula algebra S, the compatibility of a mapping C : Th (C) -+

Con (S) with the C-filters on S is the same as the compatibility of C with

the theories of C.

COROLLARY 1.9. Let (S, C) be an arbitrary logic. EC is the largest of all

operators C for C which are monotone and compatible with the theories of

C. a

The Suszko operator is set-theoretically defined through the Leibniz op

erator. It turns out that in the case of protoalgebraic logics, the Suszko

operator coincides with the Leibniz one.

THEOREM 1.10. Let (S, C) be an arbitrary logic. The following conditions are equivalent:

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200 J. Czelakowski

(i) C is protoalgebraic.

(ii) The operators Q and EC coincide, i.e., QT - ZCT for all theories TC Th(C).

(iii) QAD =CD for all S-algebras A and all filters D E Fic(A).

PROOF. (i) X (iii). Let A be an S-algebra and D E FiC(A). By The orem 1.3.(3) we have that ECD C QAD. Assume furthermore that C is protoalgebraic. Then, by Theorem 0.9.(iii), QA satisfies the condition (*)A of Theorem 1.6 above. Hence QAD C ZCD.

(iii) ? (ii). This is trivial. (ii) => (i). Assume (ii) holds. As the operator EC is always monotone,

the monotonicity of Q on Th (C) follows. Hence C is protoalgebraic. E

2. Deductive homomorphisms

Let C be a fixed logic. Let M and N be matrices which are models of C. A homomorphism h: M -> N between the two models is called C deductive (or simply deductive when C is clear from context) if, for every pair a, b of elements of M, ha = hb implies CM(a) =CM(b). Thus deductive

homomorphisms identify at most these elements a, b of the matrix M which are deductively equivalent in M, i.e., for which CM(a) = CM(b).

Not every deductive homomorphism h: M -+ N is strict and the converse is not true either: not every strict homomorphism is deductive. It is easy to see that a surjective deductive homomorphism h : M -* N is strict, where

M = (A, D), N = (B. E), iff h maps D onto E.

Suppose 1D is a congruence of the algebra reduct A of M. D is called a deductive congruence on M iff a b (mod D) implies CM(a) = CM(b), for all a, b E A. Note that every deductive congruence on M is strict. The converse need not hold. It holds for all C-models iff C is protoalgebraic. Furthermore, we observe that the Suszko congruence EM is the largest element in the poset of all deductive congruences on M.

The interdependencies between deductive homomorphisms and deductive congruences are straightforward:

PROPOSITION 2.1. Let C be a logic and let M and N be matrix models of C.

(i) Assume 1 is a congruence of the algebra of M. The canonical ho momorphism k, : M -? M/4D is deductive iff 1 is a deductive con

gruence on M.

(ii) Let h : M -* N be a homomorphism. h is deductive iff the relation kernel of h, ker(h), is a deductive congruence on M.

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The easy proof is omitted. U

We also notice that if h: M -4 N is a surjective homomorphism and

hD = E, where D and E are sets of designated values of M and N, re

spectively, then the quotient matrix M/ker(h) is isomorphic with N, cf. Proposition 0.3.(iv).

A paradigmatic example of a (surjective) deductive homomorphism is the

canonical mapping from a model M = (A, D) of C to the quotient model

M/ECM = (A/E?D, D/ECD).

PROPOSITION 2.2. Let M = (A, D) be in Mod(C). The canonical homo

morphism h: M -* M/ECM is surjective, strict and deductive.

PROOF. h is trivially surjective. We shall check it is deductive. Let a, b C A

and ha = hb, i.e., a _ b (mod ?CM). Then CA(D,a) = CA(D,b), i.e.,

CM (a) = CM (b) Clearly, a C D implies ha E D/ECM. Conversely, suppose ha E

D/ECM, i.e., a b (mod SCM) for some b C D. But then CM(a) =

CM(b) = D. So a ED. Thus h is strict.

The constitutive feature of the theory of protoalgebraic logics is that it

establishes, for each strict homomorphism between matrix models, a strict

relationship between filter lattices of these matrices. This relationship is ex

pressed in the Correspondence Property. This property cannot be extended

to a larger class of logics because it is equivalent to protoalgebraicity (Theo rem 0.9). We shall discuss below a modification of this property formulated in terms of deductive homomorphisms rather than strict homomorphisms.

This enables us to preserve strict links between filter lattices of models re

gardless of the fact the logic under consideration is protoalgebraic or not.

PROPOSITION 2.3. (The Correspondence Property for Deductive Homomor

phisms). Suppose M and N are models of a logic C. Let h: M -* N be a

deductive homomorphism and F E Fic(M). Then F = h-1hF.

PROOF. The inclusion F C h-'hF is immediate. To prove the reverse inclusion assume a c h-1hF. Then ha = hb for some b c F. Since h is

deductive, CM(a) = CM(b). But CM(b) C F. Hence a c F. U

PROPOSITION 2.4. Let M and N be models of C. Suppose h: M -* N

is a surjective and strict deductive homomorphism. Then for every filter F C Fic(M), hF is a member of Fic(N).

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202 J. Czelakowski

PROOF. It can be directly verified that generally if the inverse image of a set under a surjective mapping is a C-filter, then the set is also a C-filter. This fact and the Correspondence Property for Deductive Homomorphisms imply that hF is a deductive filter on B. Since D C F and h is strict, we have E = hD C hF. So hF E FiC(N). C

COROLLARY 2.5. Let M be a model of C and h the canonical homomorphism from M onto M/ECM. Then, for every filter F E Fic(M), hF belongs to

Fic (M/ECM).

COROLLARY 2.6. Let M and N be models of C. Suppose h: M -+ N is a surjective, strict deductive homomorphism. Then hCM(X) = CN(hX) for every subset X of M.

COROLLARY 2.7. Let M and N be models of C. Suppose h: M -+ N is a surjective, strict deductive homomorphism. Then

SCM - h-1(ECN)

and the mapping which to each filter F E Fic(M) assigns the filter hF E Fic(N) is an isomorphism between the lattices Fic(M) and Fic(N).

PROOF. The first part follows from Corollary 2.6 and the definition of SCM. The second part is also an easy consequence of Corollary 2.6. U

3. Suszko-reduced models

DEFINITION 3.1. Let M be a model of a logic C. The matrix M is called Suszko-reduced iff ZCM is the identity relation on M.

Since the Suszko congruence YCM of any model M of C is contained in the Leibniz congruence QM of this model, every Leibniz-reduced model of

C is Suszko-reduced. The following theorem justifies the use of the phrase " Suszko-reduced "in the above definition.

THEOREM 3.2. Let M = (A, D) be a model of a logic C. The following conditions are equivalent.

(i) M is Suszko-reduced; (ii) M is isomorphic with the matrix N/ECN for some model N.

PROOF. The implication (i) =S (ii) is immediate. (ii) => (i). It suffices to show that for an arbitrary model N = (A, D) of C,

the model N/ECN is a Suszko-reduced matrix. So let M := N/ECN and let

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The Suszko Operator. Part I. 203

h: N -* N/ECN be the canonical homomorphism. h is strict and deductive. Assume CM ((p(ha, hc)) = CM (o(hb, hc)) for all formulas bo(x, u,... ., un) of S and all sequences c = cl,... ,cn of elements of A. Then, by Corollary

2.6, hON(Qp(a, c)) = CM(Qo(ha, hc)) = CM(y'(hb, hc)) = hCN(f(b, c)), for all (o and c, which gives h-WhCN(p(a,c)) = h-WhCN(o(b,c)), for all f and c. Hence, by the Correspondence Property for Deductive Homomorphisms,

CN ((p(a, c)) = CN (Qo(b, c)), for all o and all c. Thus a b (mod ZCN), i.e., ha = hb. So (i) holds. C

Let M be in Mod(C). The quotient matrix M/ECM is called the Suszko-reduction of M.

The above theorem states that a matrix model of C is Suszko-reduced iff it is isomorphic with the Suszko-reduction of some model.

For any class K C Mod(C) we let Ksu(C) denote the class {M/ECM: M C K} of the Suszko-reductions of the members of K. Thus

Modsu (C)

is the class {M/ECM : M C Mod(C)}. It follows from Theorem 3.2 that I(Modsu(C)) coincides with the class of all Suszko-reduced models for C.

Since the Suszko congruence SCM of any model M of C is contained in the Leibniz congruence QM of this model, every Leibniz-reduced model of

C is Suszko-reduced. Consequently,

I(Mod* (C)) C I(Modsu(C)) .

The reverse inclusion holds iff C is protoalgebraic, see Corollary 5.5.

LSU (C)

denotes the family of Suszko-reductions of Lindenbaum models for C. Thus Lsu(C) consists of all matrices of the form (S, X)/ECX with X ranging over the set Th (C).

Because the canonical homomorphism from M to M/ECM is strict and surjective, the matrices M and M/ECM induce the same consequence op eration on S. Consequently, each logic C is complete with respect to either of the classes Modsu(C) or Lsu(C).

The following observation supplements Theorem 3.2:

PROPOSITION 3.3. Let M = (A, D) be a model of a logic C. Let T be a congruence of A such that the quotient matrix M/P = (A/T, D/4') is Suszko-reduced. Then ZCM C I.

PROOF. Suppose a -b (mod YCM) for some a, b E A. Then, by Theorem

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204 J. Czelakowski

(*) p(a, b, c) C CA(D, 4(a, b, c)) and 0 (a, b, c)) E CA(D, p(a, b, c)),

for all pairs (my) of formulas and all sequences c of elements of A. (*)

implies that in the quotient matrix M/T:

p([a], [b], [c]) C CA/I (D/' I',4([a], [b], [c])) and

4'([a], [b], [c]) C CA/E (D/@ I,(.p([a], [b], [c])),

for all pairs (y, 4') and all strings c. ([a] is the equivalence class of a with

respect to T.) Thus [a] _ [b] (mod EC/4ID/4). But M/IT is Suszko-reduced

by the assumption. Hence [a] = [b], i.e., a - b (mod A). This proves that

SCM C T. U

COROLLARY 3.4. Let M = (A, D) be a model of a logic C. SCM is the only

deductive congruence of M yielding a Suszko-reduced quotient. U

4. Determinators of the Suszko operator

Let E(x, y, a) be a set of pairs (p(x, y, u), 4(x, y, u)) of sentential formulas

of S built-up with variables x, y and possibly other variables u = u1, u2,. The variables x and y are the main variables of E(x, y, u). The variables of u

are called parametric variables, or simply parameters. The formulas in each

pair (o(x, y, U), '(x, y, u)) individually contain only finitely many parame ters. But if E(x, y, u) is infinite, the pairs belonging to it may collectively contain an infinite number of parameters and therefore u may be a sequence of type w.

Given an algebra A similar to S and a set D C A we define the binary relation EA(D) on A by the stipulation:

(AR) (a, b) C EA(D) iff CA(D, W(a, b, c)) CA(D, 4(a, b, c)) for all pairs

(po, 4) E E(x, y, u) and all strings c = c1, c2,... of elements of A.

EA(D) is called the analytical relation on A determined by the sets D

and E(x, y, u).

NOTE. It is easy to see that the analytical relation EA(D) is equivalently

characterized by the condition:

(AR)* (a,b) C EA(D) iff CA(D,(p(a,b,cl,... *cn)) = CA(D,4(a,b,cl,...

Cn)) for all pairs (p(x,y,uj,... .,un),4'(x1yu,... ,un)) C E(x,y,u) and all sequences cl, . . ., cn of elements of A.

The difference between (AR) and (AR)* is in the fact that in (AR) the values c = cli, c2, .. ., are uniformly assigned to the parameters u = u1, u2, . ... in all

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pairs (,o, A) c E(x, y, u) simultaneously while in (AR)* each pair ((p, A) C E(x, y, u) is individually evaluated in A. In the formulation of (AR) the

quantification is usually being made over infinite sequences of elements of A while in (AR)* one quantfies over finite strings of members of A. c

LEMMA 4.1. If the relation EA(D) is reflexive on A, then Y3CD C EA(D).

PROOF. Assume a _ b (mod ECD) for some a, b E A. Hence, by Theorem

1.3.(1):

(a) CA(D, p(a, b, c)) = CA(D, <p(b, b, c)) and CA(D, V) (a, b,c)) =

CA(D,? (b, b, c)) for all pairs (a, A) e E(x, y, u) and all sequences c c1, c2, ... of elements of A.

Since the relation EA(D) is reflexive, (b, b) C EA(D) which means that

(b) CA(D, y(b, b, c)) = CA(D, 4'(b, b, c)) for all (o, A) c E(x, y, u) and all

sequences c = c1i, c2, ... of elements of A.

It follows from (a) and (b) that

CA(D, 0o(a, b, c)) = CA(D, /(a, b, c)) for all pairs (p, b) C E(x, y, u) and

all sequences c = c1, c2, ... of elements of A.

So (a, b) E EA(D). A

Let M = (A,D) be a matrix model of C. A set E = E(x,y,zu) of pairs of sentential formulas is called a (parameterized) determinator for M if EA(D) = SCD. Thus E(x, y, a) is a determinator for M iff the following equivalence holds, for all pairs a, b C A:

(1) a -b (mod ECD) iff CA(D, (p(a, b, c)) = CA(D, 0 (a, b, c)) for ev ery pair (p(x, y, U), b(x, y, u)) C E(x, y, u) and every string c

c, ... ., cn of elements of A.

E(x, y, u) is a (parameterized) determinator for a class K C Mod(C) if the equality EA(D) = ZCD holds for all matrices M = (A, D) in K.

E(x, y, u) is called a (parameterized) determinator of the Suszko operator for C iff E(x, y, u) is a determinator for the Lindenbaum class L(C), that is, if the equality Es(T) = ECT holds for all theories T e Th (C).

E(x, y, u) is called a (parameterized) determinator of the extended Suszko operator for C iff E(x, y, u) is a determinator for the class Mod(C). Equiv alently, E(x, y, a) is a determinator of the extended Suszko operator for C iff the equality EA(D) = YjCD holds for all algebras A and all filters D e Fic(A). Determinators of the extended Suszko operator for C are also called global determinators.

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206 J. Czelakowski

THEOREM 4.2. For every logic C, the extended Suszko operator for C pos sesses a (possibly infinite) parameterized determinator.

PROOF. Fix two variables x and y and define E = E(x, y, u) to be the set of all pairs (S(x, u), 5(y, u)), where 6(x, u) is an arbitrary sentential formula of S built-up in x and possibly other variables u = ul, u2,... Then, for any algebra A and any filter D C FiC(A), Theorem 1.3.(1) yields:

a _ b (mod YECD) iff,

for every sentence o(xu, ..., un) of S and all cl,... , cn E A,

CA(D, (p(a, cl: ... IcC)) = CA(D (p(b, clI,... ,c)) iff,

for every pair (6 (xu),6(y,u)) e E and all c,. .. ICn E A,

CA(D, 6(a, cl v... IcC)) = CA(DI 6(b, cl: *... *cC)) iff

(a, b) C EA(D). A

Apart from the determinator defined as in the proof of the above theorem, we define here another worthwhile mentioning determinator of the extended Suszko operator for C.

PROPOSITION 4.3. Let (S, C) be an arbitrary logic. Define E' = E'(x, y, u) to be the set consisting of all pairs (so(x, y, u), b(x, y, U)) of sentential formulas such that C((p(x, x, u)) = C(O(x, x, u)). Then E' is a determinator of the extended Suszko operator for C.

PROOF. One may directly verify that E'(x, y, u) is indeed a determinator. We give a short proof of the present proposition which makes use of Lemma 4.1. Evidently, the determinator E = E(x, y, u), defined as in Theorem 4.2, is included in the set E'. Consequently, for any algebra A and any filter

D E Fic(A), the relation EA(D) is included in EA(D). Furthermore, it follows from the definition of E' that the relation EA(D) is reflexive. Hence, by Lemma 4.1, ZCD C EA(D). Thus ECD C E (D) C EA(D) = ECD which gives that E$(D) = ECD.

COROLLARY 4.4. Let E = E(x, y, u) be a parameterized determinator of the extended Suszko operator for C. For every model M = (A, D) of C the following conditions are equivalent:

(i) M is Suszko-reduced.

(ii) EA(D) is the identity relation in A.

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The Suszko Operator. Part I. 207

Let A be an S-algebra and let R be a binary relation on A. The relation R is said to be compatible with the operations of the algebra A if for every k-ary operation f of A (k > 0) and any two k-tuples (al, ... , ak), (b,... ., bk) of elements of A, the conditions (a,, b1) c R. ... , (ak, bk) C R imply that

(f(al, ..., ak),f(bl, * .., bk)) c R.

Suppose furthermore that C is a logic in S and let D be a subset of A. We say that R is compatible with C-interderivability modulo D if, for all a, b c A, (a, b) E R implies CA(D, a) = CA(D, b).

THEOREM 4.5. Let (S, C) be a logic and let M = (A, D) be a model of C. Furthermore, let E(x, y, u) be a set of pairs of S-formulas. E(x, y, u) is a parameterized determinator for M if the following three conditions are satisfied:

(pd)o The relation EA(D) is reflexive.

(pd)1 The relation EA(D) is compatible with C-interderivability modulo D.

(pd)2 The relation EA(D) is compatible with the operations of A.

PROOF. (a). Suppose E(x, y, u) is a parameterized determinator for M. Then evidently (pd)0, (pd)1 and (pd)2 are direct consequences of the formula (1) and the properties of EC (see Theorem 1.3).

(=). The proof of this implication is based on the following enhanced version of Theorem 1.6:

LEMMA 4.6. Let (S, C) be a logic. Suppose (A, D) is a matrix for S. If R is a binary relation on A satisfying the properties of compatibility with C interderivability modulo D and compatibility with the operations of A, then

RC ESD.

PROOF OF THE LEMMA.

CLAIM. If a binary relation R is compatible with the operations of A then it has the replacement property, i.e., for every pair a, b E A, (a, b) e R implies that ((p(a, el,... ,em), (b, e,...,erm)) E R for all S-sentences p(xu, ....

um) and all el,...,em E A.

The proof of the claim is by induction on complexity of p(x, u1, . .. ,um). Assume (a, b) C R. The thesis evidently holds if p(x,u1,...,um) is the variable x. Suppose p(x, u1,... , um) is a compound sentence of the form

F((pl(x, u,... ,urm) . .. . pn(x, ul... ,um)) and assume the thesis of the claim holds for the sentences 91(x ) U1 *. Um),*. i, yn(X U1, ... -urm). Let e1,.

em C A. Define:

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208 J. Czelakowski

ci :o i(a,ei,... , em),. Cn = n(a, el,... I m),

di := ((be,. . . em)) *... dn =n(b, el, I * I em)

By the inductive hypothesis we have that (clI di) C RI... , (cn Idn) E R. Hence, by the compatibility property of R with the operations of A, we ob tain that (FA(cl,... CO),FA(dl ... Idn)) E R, i.e., (fo(a,el,... ,em),Ip(b,el,

,em)) E R. This proves the claim. To prove the lemma, assume (a, b) E R for some a, b E A. As R is compat

ible with the operations of A, the above claim implies that (p(a, el,... I cm), (p(b,el,... ,em)) E R, for for all S-sentences (xulI,... ,Ium) and all el,...I em C A. In turn, since R is compatible with C-interderivability modulo

DI we get that CA(D, p(a, e,... ,em)) = CA(D, (b, el, ... , em)), for all S sentences W(xu 1,... , UM) and all ei,...,em C A. Hence a _ b (mod SAD), by Theorem 1.3.(1). So the lemma holds. c

To conclude the proof of Theorem 4.5, assume the relation EA(D) satis fies (pd)o - (pd)2. Since EA(D) is reflexive, we have that SCD C EA(D) by

Lemma 4.1. In turn, Lemma 4.6 yields EA(D) C ZCD. So EA(D) = SAD. This means that E(x, y, u) is a determinator for M.

This proves the theorem. c

COROLLARY 4.7. Let (S,0) be a logic and let E(x,y,u) be a set of pairs of S-formulas.

(1) E(x, y, u) is a parameterized determinator of the Suszko operator for C iff the conditions (pd)o - (pd)2 hold for every Lindenbaum matrix in L(C).

(2) E(x, y, u) is a parameterized determinator of the extended Suszko operator for C iff the conditions (pd)o - (pd)2 hold for every matrix in Mod(C).

A common problem besetting both the theories of protoalgebraic logics and non-protoalgebraic logics is that of giving an adequate account of the properties that transfer from the theory lattice to arbitrary filter-lattices. We mention here one such a property. Corollary 4.7.(2) provides a global char acterization of determinators of the extended Suszko operator in the sense that it requires the clauses (pd)o - (pd)2 to hold for all filter lattices. We ask whether from the fact that (pd)o - (pd)2 hold only for the theories of C only one can deduce that these conditions transfer to the lattices of C-filters on arbitrary S-algebras. In other words, we ask whether the fact that E is a determinator for the Suszko operator for C implies that E is a determinator of the extended Suszko operator for C. We shall discuss this problem in ?

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The Suszko Operator. Part I. 209

6. Here we note the following observation which states the invariability of determinators under surjective, strict and deductive homomorphisms.

THEOREM 4.8. Let (S,0) be an arbitrary logic and let E(x,y,u) be a set of pairs of sentential formulas of S. Furthermore, let h be a surjective, strict and deductive homomorphism from a matrix M = (A, D) to a matrix

M' = (A',D'), where M and M' are models for C. Then E(x,y,u) is a determinator for M iff it is a determinator for M'.

PROOF. We have to prove that:

(*) EA(D) = SAD iff EA'(D') - SAND'*

Evidently hD = D'. Furthermore SAD - h-l(ECD') by Corollary 2.7.

We need the following observation:

LEMMA 4.9. Under the assumptions of the above theorem,

EA (D) = h- 1(EA' (D')).

PROOF OF THE LEMMA. Assume that a, b E A and (a, b) E EA(D), i.e.,

CM((p(a, b,c)) = CM(0 (a, b,c))

for every pair (o(x, y, u), + (x, y, u)) E E(x, y, u) and every string c = cl, ...,

Cn of elements of A. Then, since h(D) = D',

(1) CM' (h((p(a, b, c))) = CM' (h (04(a, b, c)))

for every pair (, (x, y, u), 0 (x, y, u)) C E(x, y, u) and every string c = cl, ....

Cn of elements of A. Since h is surjective, (1) gives that

CQ(o(pha, hb, c')) = CQ(O (ha, hb, c'))

for every pair (p(x, y, u), 4(x, y, u)) E E(x, y, u) and every string c' = cl, .... c' of elements of A'. This shows that (ha, hb) C EA$(D'), proving that (ab) C h1(EAI(D')). Thus EA(D) C h (EA(Dl)).

Conversely, assume that (a, b) e h-1(EA'(D')). So (ha, hb) E EA,(D') which means that

(2) CM' (o(ha, hb, c')) = CM' (Qe(ha, hb, c'))

for every pair (p(x, y, u), (x, y, u)) E E(x, y, u) and every string c' = cl

c' of elements of A'. But the surjectivity of h and (2) imply that

(3) CM' (h (p(a, b, g))) = CM' (h(4 (a, b, c)))

for every pair ('p(x, y, u), (x, y, u)) C E(x, y, u) and every string c = cl, ....

Cn of elements of A.

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210 J. Czelakowski

Corollary 2.6 gives that

(4) CM' (h((p(a, b, c))) = hCM (p(a, b, c)) and CM' (h(4 (a, b, c))) = hCM (0 (a, b, c))

for all pairs (fo(x, y, u), (x, y, U)) E E(x, y, u) and all sequences c = cl, ...

cn of elements of A. Furthermore, by Proposition 2.3,

(5) CM((o(a, b,c)) = h-1hCM(p (a, b, c)) and CM(4'(a, b, c)) =

h-1hCM(4 (a, b,c))

for all pairs (Wo(x, y, u), 4(x, y, u)) E E(x, y, U) and all sequences c = cl, ...

cn of elements of A. Taking into account (3),(4) and (5), we get that

CM((p(a, b, c)) = h-1 hCM(W(a, b, c)) = h1-CM'(h(( (a, b, c))) =

h-1CM (h(0 (a, b, c))) = h-1hCM(V/ (a, b, c)) = CM(V/(a, b, c))

for all pairs (W (x, y, u), ' (x, y, u)) c E(x, y, u) and all sequences _ = cl,

cn of elements of A. Consequently,

CM(Wo(a, b, c)) = CM(4 (a, b, c))

for all pairs (Wo(x, y, u), /(x, y, u)) E E(x, y, u) and all sequences c = cl, ...,

cn of elements of A. This means that (a, b) e EA(D). So h- '(EA' (D')) C

EA (D). This concludes the proof of the lemma. U The theorem easily follows from the above facts. To prove (*) assume

first that EA(D) = YCD. Hence

(6) h-1(EA'(D')) = h-(ECSD')

by the equality ZCD - h-'(EYC,D') and Lemma 4.9. But (6) readily implies

that EA'(D') = -C/D/.

To prove the reverse implication, assume that EA1 (D') = EC, D'. Apply ing Lemma 4.9 and the equality ZCD = h-1(EC,D') we obtain EA(D) = h-1(EAi(D')) = h1(EYCD') = ZCD. So EA(D) = EQD.

COROLLARY 4.10. Let (S, C) be a logic and let E(xY, u) be a set of pairs

of formulas of S. E(x, y, U) is a parameterized determinator of the Suszko

operator for C iff E(x, y, U) is a determinator for the class LSU(C).

5. Structural properties of the class ModSu(C)

We begin with the analysis of the relationship between countable Suszko reduced models and the Suszko-reductions of Lindenbaum models.

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The Suszko Operator. Part I. 211

PROPOSITION 5.1. Let M = (A, D) be a countable Suszko-reduced model of C. Then there exists a theory X C Th (C) and a strict homomorphism g from (S, X)/ECX onto M.

The above proposition states that every countable Suszko-reduced model is a strict homomorphic image of the Suszko-reduction of a Lindenbaum

matrix.

PROOF. Let h: S -- A be an arbitrary surjective homomorphism. Let X

h-1(D) and N := (S,X)/ECX. We then define the mapping g: N -+ M

by g([a]) := ha, for all a c S. ([a] is the equivalence class of a relative to

Ex.) Let E(x, y, u) be a fixed parameterized determinator of the Suszko oper

ator for C.

CLAIM 1. g is well-defined. PROOF OF THE CLAIM. Let [a] = [/3], i.e., C(X,p(a,,3<y)) C(X, (a,/3, -y)), for all pairs (py, b) E E(x, y, u) and all strings ay of elements of S.

Then CM(hC(X, W (a, /, -y))) = CM(hC(X, (4a, /3, ny))), for all pairs (p, 4) C

E(x, y, u) and all y. Applying Proposition 0.4.(v) we obtain that CM(fp(ha,

h/, h7y)) = CM(0 (ha, ho, hey)), for all pairs (p, 7p) C E(x, y, u) and all strings -y. As h is surjective, we have that CM(p (ha, ho, c)) = CmM(4 (ha, ho, c)), for all pairs (p, A) E E(x, y, u) and strings c of elements of A. Hence ha - ho (mod SCM). Since M is Suszko-reduced, ha = ho.

CLAIM 2. g is a homomorphism of the algebra S/ECX onto A. PROOF OF THE CLAIM. Let ? be an n-ary connective of S and Ca1,... ,an arbitrary sentences of S. Then g(?N([al], .. ., [ae])) = g([?(ai,..., asn)]) =

h(f(aci,. . .. an)) ?M(ha, ... han) = ?M(g([a]),. . gan])

CLAIM 3. g: N 3 M is strict. PROOF OF THE CLAIM. For a e S we have: [a] e X/EX iff [a] = [/] for some 3 C X iff a C X iff ha C D.

This concludes the proof of the proposition. C

NOTE. The homomorphism g defined in Proposition 5.1 need not be de ductive. For let X be an arbitrary theory of C and M := (S, X)/QX,

N := (S, X)/ECX. If the homomorphism 9 : S/ECX - S/QX, de fined by g([a]) = [a]Qx, a C S, were deductive, we would have that ECX = QX which need not hold unless C is protoalgebraic. Indeed, sup pose a =_ / (mod QX) for some a, /3. This implies that p(x/a) _ (x//) (mod QX) for all sentential formulas bo C S and all variables x C Var (p). So g([k(x/a)]) = g([k(x//)]), for all o C S and all x. Since g is assumed

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212 J. Czelakowski

to be deductive, we then have that CN([p(x/a)]) = CN(Rp(x//3]) for all (O and x. This implies that C(X, ~o(x/a)) = C(X, p(x/l)), for all o and x. So a = / (mod ZCX). Thus QX C YCX. C

If K is a class of matrices, then Hs(K) denotes the class of all strict homomorphic images of members of K.

Proposition 5.1 as well as other results of this section give rise to the question as whether the classes I(Lsu(C)) and I(Modsu(C)) are closed under strict homomorphic images. The answer is positive for protoalge braic logics, as then Suszko-reduced models are the ordinary reduced mod els, which are always (trivially) closed under strict homomorphic images. But, in the non-protoalgebraic case, the answer to this question is negative. The following example is due to Pigozzi. Let S consist of four constants 0, 1, 2,3 and no connectives of positive rank. Let C be the deductive sys tem with one axiom 0 and the following three inference rules: { 1 }/0, {2}/1, and {3}/2. C is not protoalgebraic because only the inconsistent and the almost inconsistent logics can be protoalgebraic in this language. Let A be the algebra with universe {0, 1, 2, 3} and M := (A, {0}). The C-filters of M are {0}, {0, 1}, {0, 1, 2}, and {0, 1, 2,3}. Clearly M is Suszko-reduced. Let b be the congruence on A that identifies the elements 1 and 3 and no oth ers. Then A/ib {{0}, {1, 3}, {2}} and A/1D has just two C-filters, {{0}} and {{0},{1,3},{2}}. Let N := M/14 = (A/1,{{0}}). The congruence ZCN = EC/,{J{?} } identifies the two elements { 1, 3} and {2} . Thus N is evidently a strict homomorphic image of M but it is not Suszko-reduced. This example shows that I(Modsu(C)) is not closed under strict homomor phic images.

It follows from this example that also the class I(Lsu(C)) is neither closed under the formation of strict homomorphic images. Indeed, suppose I(Lsu(C)) is closed with respect to Hs. Hence, by Proposition 5.1, the

matrix M defined as above belongs to I(Lsu(C)). As the matrix N is a strict

homomorphic image of M, it also belongs to I(LSu(C)). A contradiction, because N is not Suszko-reduced.

We note here the following positive (and trivial) fact which follows from

Corollary 3.4:

The classes I(LSu(C)) and I(Modsu(C)) are closed under deductive strict homomorphic images.

(This fact is trivial because if M is a Suszko-reduced model of C then the identity relation is the only deductive congruence on M and hence M is its only, up to isomorphism, deductive and strict homomorphic image.)

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The Suszko Operator. Part I. 213

The fact that the extended Suszko operator is monotone enables to derive the following theorem.

THEOREM 5.2. For every logic C, the class I(Modsu(C)) is closed under the formation of subdirect products.

PROOF. Let E(x, y, u) be a fixed parameterized determinator of the ex tended Suszko operator for C. Suppose Mi = (Ai, Di) (i E I) is a non-empty family of Suszko-reduced models for C and M = (A, D) CSD I We

want to show M is Suszko-reduced as well. Let a = (ai: i C I) and b = (bi: i C I) be elements of A such that a = b (mod SCM). Hence

(l) Cm (9o(a, b, cl, .. I *CvOn)

- Cm (0(aj b, cl , ..

*COe)):

for all pairs ( E, A) E E(x, y, u) and all strings cl,... , cn of elements of A. Let 7ri be the projection of A onto Ai, i.e., ri(f) := f (i) for all f E A. i

is clearly a surjective homomorphism from M to (Ai, Di). Denote by Ci(X) the least C-filter on Mi that includes X. By Proposition 0.4.(v), for each

Ci (riCm(p(a, b, cl, COc))) = Ci((o(ai, bi, cl (i), , Cn()

and

Ci (ri Cm ( (a, b, cl, .. , CO)))

- Ci (7 (ai, bi, cl (i, ,n n(M))

for all pairs (m, y) C E(x, y, u) and all cl,... ,cn C A. On the strength of (1), the last equations give that for any i E I,

(2) Ci(p(ai,b i(i)). . . ,cn(i))) = Ci('(ai, bi, cl(i), .v . ()))_

for all pairs (p, 4) c E(x, y, a) and all cl, ... , Cn ,E A.

As 7ri is onto, (2) yields

Ci ((p(ai, bi , cl , ... * CO)) = Ci (O(ai , bi, Cl , ...

CO))

for all i c I, all pairs (p, y) C E(x,y,u) and cl,... ,c CE Ai. Thus ai-bi (mod YCMi), for all i C I. Hence, since Mi is Suszko-reduced, ai = bi, for all i E I, i.e., a = b. This proves that M belongs to I(Modsu(C)). U

Given a class K of matrices we let Ps(K) denote the class of all isomor phic copies of subdirect products of members of K.

The following theorem establishes the relationship between the class of Suszko-reduced models and the class of Leibniz-reduced models of an arbi trary logic C. According to it, every Suszko-reduced model for C is isomor phic with a subdirect product of a family of Leibniz-reduced models.

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214 J. Czelakowski

THEOREM 5.3. For every logic C, I(ModSu(C)) = PS(Mod*(C)).

PROOF. Since Mod*(C) C I(ModSu(C)), Theorem 5.2 implies that

Ps(Mod*(C)) C I(Modsu(C)). To prove the reverse inclusion, we as sume that M = (A, D) is a Suszko-reduced model of C. By the definition, there exists a family {Di: i E I} C FiC(M) of deductive filters such that

(*) IdA = SCM - F{QADi: i C I}.

Define Mi := (A,Di) for each i E I. Evidently, the quotient matrices

Mi/QMi belong to Mod*(C). If a C A, then [a]i is the equivalence class of a with respect to QMj. Hence, by (*), the mapping h given by h(a) := ([a]i: i c I), for all a c A, establishes an isomorphism between the matrix M and a subdirect product of the matrices Mj/QMj, i C I. So M e PS(Mod*(C)).

By a similar argument we get the following inclusion:

COROLLARY 5.4. For every logic C, I(Lsu(C)) C Ps(L*(C)).

The known result that C is protoalgebraic iff Mod*(C) is closed under subdirect products, see [3], is deducible from the above theorems:

COROLLARY 5.5. Let C be an arbitrary logic. The following conditions are equivalent:

(1) C is protoalgebraic.

(2) Mod* (C) = ModSu (C)

(3) I(Mod*(C)) is closed under the formation of subdirect products.

PROOF. (1) > (2). Assume C is protoalgebraic. By Theorem 1.1O.(iii), QM = SCM in every model M of C. Hence Mod*(C) - Modsu(C).

(2) = (3). Assume (2). Hence evidently I(Mod*(C)) = I(Modsu(C)). Then apply Theorem 5.2.

(3) #> (1). Assume (3) holds. Then I(Mod*(C)) = I(Modsu(C)) by Theorem 5.3. This equality implies that for every model M of C, the Suszko reduction M/ECM is Leibniz-reduced. Hence, as can be easily checked, ECM = QM in every model M. Consequently, QAD = ZCD for all algebras A and all filters D C Fic(A). Then apply Theorem 1.10. U

Given a logic C we define Alg(C)

to be the class of the algebra reducts of Suszko-reduced models of C, i.e.,

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The Suszko Operator. Part I. 215

Alg(C) : {A: (A, D) E I(Modsu(C)) for some D C A}.

The following observation is due to Font and Jansana;

THEOREM 5.6. For every algebra A the following conditions are equivalent:

(i) A E Alg(C);

(ii) f{QAD: D C Fic(A)} = IdA.

PROOF. (i) => (ii). Assume A c Alg(C), i.e., (ADo) C I(ModSu(C)) for

some Do C A. Hence IdA - cDo - {QAD: D E Fic(A) & Do C D} D

n{QAD: D C Fic(A)} D MdA which gives that n{QAD: D E Fic(A)} IdA. So (ii) holds.

(ii) =X (i). Assume (ii). Given an algebra A define Do:= Fic(A). Do is the least deductive filter on A. Then ?CDo - F{AD D E Fic(A)} =

IdA. So (A, Do) is Suszko-reduced and therefore A C Alg(C). c

The above result establishes the basic relationship between the theory of the Suszko operator and the general semantic approach to sentential logics based on the notion of an abstract logic and developped in [15]. This ap proach was initiated by Suszko himself and his collaborators in the sixties, see [4]. An abstract logic is a pair L = (A, C), where A is an algebra and C is a closure system on A. Of particular significance are abstract logics of the form (A, Fi C(A)), where C is a fixed deductive system and A is an arbitrary algebra. The crucial notion in Font-Jansana's theory is that of a reduced abstract logic. An abstract logic L = (A, C) is reduced if the congruence

flQAD: D e C}, which is called the Tarski congruence, is the identity relation. Theorem 5.6 thus states that an algebra A is the algebra reduct of a Suszko-reduced model iff the abstract logic (A, Fi C(A)) is reduced in Font-Jansana's sense. As a consequence, the class they denote by Alg(C) concide with ours. This fact enables one to apply the methods worked out for abstract logics in the theory of the Suszko operator.

It follows from Theorem 5.2 that Alg(C) is closed under the formation of subdirect products. But, -in the presence of Theorem 5.6, this fact was essentially proved by Font and Jansana [15], see the remarks following their Theorem 2.23. Also Theorem 5.3 is strictly related to their Theorem 2.23.

If K is a class of matrices, then S(K), P,-R(K), and PR(K) denote, respectively, the class of all isomorphic copies of submatrices of the matrices of K, the class of all isomorphic copies of u-reduced products of families of K-matrices, and the class of isomorphic copies of reduced products of families of K-matrices.

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216 J. Czelakowski

THEOREM 5.7. For every logic C, I(ModSu(C)) C HsSPoJR(LSu(C)). Furthermore, if C is finitary, then I(Modsu(C)) C HSSPR(LSu(C)).

PROOF. The proof of the first part of the theorem is based on the following facts:

(1). For any class K of matrices, HsSP,-R(K) is the least class of matrices that includes K and is closed under the operations Hs, S and P(,-R. (2). If a class K of matrices is adequate for a logic C, then Mod*(C) C

HsSP,-R(K).

For the proof, see [7, Theorem 0.6.1]. It follows from (1) that the class HsSP,-R(K) is closed under the

formation of subdirect products because it is closed under direct products and submatrices.

As Lsu(C) is adequate for C, (2) implies that Mod*(C) C HSSP,-R (LSu(C)). Hence, by Theorem 5.3, I(ModSu(C)) = PS(Mod*(C)) C

Ps(HsSPo,-R(LSu(C))) = HsSP,-_R(LSU(C)). This completes the proof of the first statement.

Let us note that the class HSSP,-R(LSu(C)) is contained in Mod(C) since the class of models of each logic is closed under the operations Hs, S and P,-R, see e.g. [7, Theorem 0.6.1].

The proof of the second part is fully analogous to the proof of the first part. It is based on the following observations: (3). For any class K of matrices, HSSPu(K) is the least class of matrices that includes K and is closed under the operations Hs, S and Pu. (4). If a class K of matrices is adequate for a finitary logic C, then Mod* (C)

C HSSPR(K).

For the proof of (3) and (4), see also [7, Theorem 0.6.1].

6. Natural extensions and proper Suszko-reduced matrices

It is sometimes convenient (see [10]), to view each sentential logic (S, C) more broadly as a class of sentential calculi (S', C'), where S' is an absolutely free algebra of the same signature as S and C' is a consequence operation in S'

defined in a certain cannonical way. Intuitively, (S', C') is the "same "logic as (S, C), i.e., both C and C' have the same schemes of rules of inference. The only difference is that C' operates on a language which has more variables than the language S. Each such a pair (S', C') is called a natural extension of the logic (S, C). Formally, the notion of a natural extension of a logic is defined in the following way.

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Let S be a sentential language. A sentential language S' is called a prolongation of S if S and S' have the same (countable) set of connectives but S' has more variables than S, i.e., Var(S) C Var(S').

DEFINITION 6.1. (LoS and Suszko [21], Shoesmith and Smiley [26]. Let (SC) be a sentential calculus. A sentential calculus (S',C') is called a natural extension of (S, C) if the following conditions hold:

(a) S' is a prolongation of S,

(b) C'(X) = U{hC(h-1X) : h E Hom(S,S')}, for every set X C S'.

We will be mainly concerned with natural extensions of sentential logics

(S, C). The language S is then countably infinite. Each logic is a natural extension of itself. It is not difficult to show that (S', C') is a natural exten sion of (S ,) iff S' is a prolongation of S, C(X) = S n C'(X) for all sets

X C S. and both C and C' have the same cardinality. It follows that if S' is a prolongation of S, then there exists exactly one consequence operation C' on S' such that (S', C') is a natural extension of the logic (S, C). The con sequence C' in S' is determined by the rules which have the same schemes as the rules of the logic C. Consequently, for any algebra A similar to S, the families of C-filters and of C'-filters on A coincide. Equivalently, C' is determined by the class Mod(C). So C and its any natural extension C' have the same model class, i.e., Mod(C') = Mod(C).

We observe that, in particular, if (S', C') is a natural extension of (S, C) then Th(C') = Fic(S'), i.e., the family of C'-theories coincides with the family of C-filters on S'.

It follows from the above definition that (S', C') is a natural extension of (SI C) iff

Th(C')= n {X C S': h CX Th(C)}. he Hom (S,S')

The following characterization of natural extensions is essentially due to Los and Suszko [21]:

THEOREM 6.2. Let (S, C) be a sentential logic. Let S' be a prolongation of S. Define the operation C' : P(S') -* P(S') as follows: for X C S' and

a E S',

a E C'(X) iff there exists a denumerable set Y C X and an automorphism e of S' such that ea c S. eY C S and ea C C(eY).

Then (S', C') is a natural extension of (S, C). o

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218 J. Czelakowski

We note that each automorphism of a sentential language is uniquely determined by a permutation of the set of variables of this language.

Given a sentential logic (S, C), we define the extended Lindenbaum class for C, in symbols:

ExL(C)

to be the class being the union of the Lindenbaum families L(C'), where (S', C') ranges over all possible natural extensions of (S, C). ExL(C) thus contains matrices whose algebra reducts are absolutely free algebras of arbi trarily high cardinalities. Clearly, ExL(C) C Mod(C). The class ExLSu (C) of Suszko-reductions of the matrices of ExL(C) is therefore a subclass of

Modsu(C). The members of ExLSu(C) are called Suszko-reductions of ex tended Lindenbaum matrices.

The members of the class I(ExLSU(C)) are called proper Suszko-reduced matrices. We use a special symbol

Modsu(C)

to denote the class of proper Suszko-reduced matrices. Thus Modsu(C) =

I(ExLsu (C)). The following theorem states that every Suszko-reduced model is a strict

homomorphic image of a proper Suszko-reduced model.

THEOREM 6.3. For any logic (S, C), I(ModSu(C)) CIHs(Modsu(C)).

PROOF. It suffices to prove the following lemma:

LEMMA 6.4. Let M = (A, D) be an arbitrary Suszko-reduced model of C. Let (S', C') be a natural extension of (S, C) such that IAI < IS'J. Then there exists a theory X E Th (C') and a strict homomorphism g from (S', X)/ECX onto M.

To prove the lemma, repeat the argument presented in the proof of Propo sition 5.1. M

The next theorem states that the property of being a determinator trans fers from the Lindenbaum family L(C) to the extended Lindenbaum family

ExL(C) and hence to the class Modsu(C).

THEOREM 6.5. Let (S, C) be an arbitrary logic. Let us assume that E(x, y, u) is a determinator of the Suszko operator for C. Then E(x, y, u) is a deter

minator for the class Modsu(C).

PROOF. In view of Theorem 4.8 it suffices to prove the following fact:

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The Suszko Operator. Part I. 219

THEOREM 6.5*. Let (S, C) be a logic and let (S', C') be a natural exten sion of (S, C). If E(x, y, u) is a determinator for L(C), then it is also a determinator for the family L(C').

Under the assumptions of Theorem 6.5*, we first prove the following lemma:

LEMMA 6.6. Assume X E Th(C'), k C w, k > 1, a1, . ... ,ak E S, /1, ... ,3k

E S' and

(1) C'(X, p(ai, /3,)) = C'(X, 0 (ai, ,))

for i = 1, . . . , k, for all pairs (I, b) E E(x, y, u) and for all sequences -y of

elements of S', where the length of y is the same as the length of u. Then for each sequence 6 of elements of S' of the length of u there exists a countable language Se, and a set X, of formulas such that:

S C SW C S. all,.... ,aOk C SWf 1, ...

,k E Sw,

Sw contains the formulas of 6,

x. c So n x,

(2) C'(X<, Wp(ai, S3i -y)) = C'(Xw, (a?i, 3j, -y))

for i = 1, ... , k, for all pairs (I, 4) E E(x, y, u) and for all sequences a of elements of S, of the length of u.

PROOF OF THE LEMMA. In order to simplify the notation, we assume that E(x, y, u) contains only one parameter, say u and that k = 1. (In the general case the proof is an easy modification of the reasoning presented below. Also the Note following the definition of a determinator is relevant here.) So, for some pair a, / C S' we have that

(1)* C'(X, '(a, /3, )) = C'(X, 4'(a, 3, y))

for all pairs (a, 4) C E(x, y, u) and for all -y C S'. Let 6 e S'.

We inductively construct an w-chain of countable sublanguages of S',

So C SI C ...

and a chain

X0 C_ Xi C ...

of subsets of X such that a, 3 ,6 E S1, and, for all n C W, Xn C Sn+1 and

C'(Xn, (p(Of, /, -)) = C'(Xn (a, /3 y))

for all pairs (a, A) c E(x, y, u) and for all -y C Sn.

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220 J. Czelakowski

We put So :- S. As S and E(x, y, u) are countable and card (C')

card (C), there exists a countable set Xo C X such that

C' (XO, WO(a, A, a) ) = C' (XO, 0 (a, A, a))

for all pairs (p, A)) c E(x, y, u) and for all ay c So. The set Xo need not be

a subset of So. We define S, to be the sublanguage of S' generated by the

variables of So, the variables occurring in a, 3, 6, and the variables of Xo.

Clearly S, is countable. Suppose that for n > 1, the countable sublanguages So C S, C ... C S,,

of S' and the countable sets Xo C X1 C ... C Xn have been defined so that

X0 C S1,...,X. l C S, and

(i)n C'(Xn, (p(a,3, )7a) =

C'(Xn, '(a,/ Ad y))

for all pairs (Qp, o) C E(x, y, u) and for all -y E Sn. Since Xn need not be a

subset of Sn, we define Sn+i to be the sublanguage of S' generated by the

variables of Sn and those occurring in the formulas of X,. Clearly Sn+I is

countable. Hence, by (1)n, there exists a countable set Xn+j C S' such that

Mln+I C (Xn+l, O(a, 0, a)a) = C (Xn+l, O(al0g d) M))

for all pairs (~ok0) C E(x,y,u) and for all -y E S,+,. The sets Sn+j and

Xn+j are therefore defined. We set: S, := Unix Sn and X, := Unfi Xn. Thus: S, is a countable

sublanguage of S', oa, 3, 6 C SG,, X, is a (countable) subset of S, and

C'(Xw, Wp(a, 3, ay)) = C'(Xg,, (ez,a, ay))

for all pairs (p,) e E(x,y,u) and for all ty c S,.

We now pass to the proof of Theorem 6.5*. We apply Theorem 4.5 to

verify that E(x, y, u) satisfies conditions (pd)o - (pd)2 for L(C').

(pd)o holds for the matrices of L(C') because for each pair (opt) e

E(x, y, u), y9(x, x, u)/I(x, x, u) and (x, x, u)/p(x, x, u) are rules of C and

hence they are also rules of C'. As to (pd)1, assume X C Th(C'), a,/ E S' and

C'(X, p(a, /3, ( )) = C'(X, b(SO, /, -y))

for all pairs (p, A) c E(x, y, u) and all sequences My of elements of S', where

the length of each _ is the same as the length of u. By Lemma 6.6, there

exist a countable language SW, such that S C S, C S' and a, / c SW, a set

XW c X n S, so that

(1)W C'(XW W(O, A, /y)) = C'(XW, b (a,R , / ))

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The Suszko Operator. Part I. 221

for all pairs (~o, k) E E(x, y, u) and for all sequences -y of formulas of S,. Let C, be the restriction of the consequence operation C' to the language

S. Thus CW(Y) := SWnC'(Y) for all Y C S,. It is easy to see that (SW, OCW) is a natural extension (S, ) and (S', C') is a natural extension of (SW, CW). Since S, is countable and hence isomorphic with S, we see that (S, O CW) is an "isomorphic copy "of (S, C) in the obvious sense. This implies that

E(x, y, u) is also a determinator for L(CW). But (1)W implies that

CW(XW, <( a, , /y)) = Cw(Xw,(A,/3y))

for all pairs ((p,) C E(x, y, u) and for all sequences -y of formulas of SW. As E(x, y, u) satisfies (pd)1 for the matrix (SW, CW (XW )), we obtain that

CW(XW:, a) = CW(XW, /). Hence, as (S', C') is a natural extension of (SW, CW), it follows that C'(XW, a) = C'(XW, /). Since XW C X, we thus get that

C'(X, a) = C'(X, 3). So (pd)1 holds for (S', X). As to (pd)2, assume X c Th(CO), a1, . . . ,ak E S/, 1, . . . , Ok C S and

let f be an k-ary connective. Furthermore, let

C'(Xp(ai, 3i y)) = C'(X, (ai, 3ily))

for i = 1,...,k, for all pairs (ps,+) E E(x, y, u) and for all sequences -y of S'-formulas.

Let 6 be an arbitrary sequence of S' of the same length as that of u. We have to show that

(2) C'(Xp ((f(al, .. ., ,ak),f (/3i, ... .,3k),1)) = C'(X, b(f(al, ... , ak),

f (pl * ..

*, Ok) , 6))

for all pairs (a, A) C E(x, y, a).

By Lemma 6.6 there exist a countable language S,, and a set XW, of formulas such that S C S,, C S', al,... , ak e SuW 13, l... /3k E SW, SW

contains the formulas of 6, XW C S,, n X' and

(3) C'(Xw, o (ai,/ y)) = C'(Xw, b(ai,3i,/-y))

for i = 1,... , k, for all pairs (py, 7s) C E(x, y, u) and for all sequences -y of elements of SW, of the length of u. Define O, as above. (3) implies that

Cw(XW, <(ai, i3 -y)) = O(Xw, (ai, 3p, y))

for i = 1,... ,k, for all pairs (pO, 0) C E(x,y,u) and for all sequences -y of elements of SW, of the length of u. Repeating the argument presented in the above, first part of the proof of the theorem for the matrix (SW, Cw(X,)) and using the fact that E(x, y, u) satisfies (pd)2 for (SW, O,(XW,)), we get that

COw(Xw, p (f (al,. . *ak), (131,... * 3,k), 7)) =

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222 J. Czelakowski

Cwo(Xw:, ~ (f (as **ak), a (01, * * *, Ok,

for all pairs (my, b) E E(x, y, u) and for all sequences -y of elements of S, of

the length of u. In particular, taking the sequence a (whose elements belong

to Sw) we obtain that

C(Xw (f(a,, * *k), , k), )) =

Q, (Xw,' (f (el, * ** k), f (:l **, Ok), )

for all pairs (a, b) c E(x, y, u). Hence, as (S', C') is a natural extension of

(SaoCo), it follows that

C'(Xw, fO(f (ce, *** k) , f (:l,* *, Ak), 6))=

C'(Xw, (f(a,. ... *ak), f(/31 .... , Ok), 6))

for all pairs (p,', ) e E(x, y, u). Thus, as X, C X,

C (X, 9P(f (Ce, *O** k), f (01, * * * , Ok) , I)) C'(X, (f(a, . . *ak), f (/31,... ,k), ))

for all pairs (p, A) G E(x, y, u). So (2) holds. This proves that E(x, y, u) is

a determinator for the matrices of L(C'). c

As yet not much can be said about structural properties of the class

ModsU(C). A more thorough discussion of this problem is postponed to

Part II.

7. Continuity of the Suszko operator

Let C be a finitary logic. We say that the Suszko operator EC is continuous on Fic(A) if

EZU{Di :i c I} = U{EDi: i C I},

for every chain {Di: i E I} of C-filters on A. [Note that, by finitarity, the

union U{Di: i E I} belongs to Fic(A).] Assuming the Axiom of Choice, it is easy to see that EC is continuous

on Fic(A) iff the above equality holds for every (upward) directed family

{Di :i I} of C-filters on A.

In the standard way the notion of the direct limit of a directed system of

matrices

E = (I,{Mi: i C I},{fij: i < j,ij C I})

is defined. If the homomorphisms fij are surjective, the system is called

su jective.

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The Suszko Operator. Part I. 223

Ls(K) stands for the class of all isomorphic copies of direct limits of surjective directed systems of matrices of K.

THEOREM 7.1. Let C be a finitary logic. Suppose that the extended Suszko operator for C is continuous on Fic(A), for all algebras A. Then

Ls(I(ModSu(C))) C Hs(Modsu(C).)

The above theorem states that, up to isomorphism, the direct limit of each surjective directed system of Suszko-reduced models is a strict homo

morphic image of a Suszko-reduced model.

PROOF. Let

E = (I, {Mi: i E I}, {fij: i < ji,j e I})

be a surjective directed system of members of I(ModSu(C)). We may as sume without loss of generality that I has the least element. We may there fore also assume that the models Mi and the homomorphisms fij take the following form:

Mi := (Al Di)/4bi (i E I):

where A is a fixed algebra, Di C Fi C (A) and 1i is a strict congruence of

(A, Di);

Di C D. and JPi C (Dj for i < j;

fij([a]i) = [a]j, for all a C Ai and all i < j.

([a]i is the equivalence class of a relative to (Di.) Since Mi is Suszko-reduced, Proposition 3.3 implies that ECDi C 4)i, for

all i e I.

Let D,, := UiEI Di and (I, := UiEI bi. Then clearly D, is a deductive

filter on A and Ibo is a congruence of the matrix (A, D,). The matrix Moo :_ (A, Do)/I1o is thus the direct limit of the system E.

Let Ni := (A, Di)/EcDi, for all i C I. Since YjDi C ZCDj whenever

i < j, the mapping gij : Ni -+ Nj (i < j) given by gij([a]zE) := [a]3j, for all

a E A, is a well-defined surjective homomorphism. ([a]E, is the equivalence class of a relative to YSDi.) Therefore

(I, {Ni : i C I}, {gij : i < j, i, j C I})

is a surjective directed system of members of ModSu(C). We have YcDc, = UiE1 YCDj by the continuity of the operator EC.

This implies that the matrix Nz, := (A, Dc)/ECD,; is the direct limit of the system I'. But YjCDOO C ( ,, which entails that Moo is a strict homomorphic image of the model N,,. Since Nc, is Suszko-reduced, we infer that Moo C Hs(ModSu(C)). A

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224 J. Czelakowski

THEOREM 7.2. Suppose I(ModSu(C)) is closed under the formation of direct limits of directed surjective systems of Suszko-reduced models, i.e., Ls(ModSu(C)) C I(Modsu(C)). Then the operator EC is continuous on

Fic(A), for every algebra A.

PROOF. We begin with the following general remark: if a class K of matri ces is closed under the formation of subdirect products and direct limits of surjective systems of members of K, then K is closed under the formation of ultraproducts. Furthermore, the logic induced by any class of matri ces closed under ultraproducts is finitary, see e.g. [7, Theorem 3.2.10 and Corollary 0.4.6]. Since, by Theorem 5.2, the class I(Modsu(C)) is closed under subdirect products, the hypothesis of the present theorem implies that I(ModSu(C)) is closed under ultraproducts. Hence, as C is complete with respect to ModSu(C), it follows that C is finitary.

Let {Di : i E I} be a directed system of C-filters on an algebra A. We wish to show C U{D i: i If = U{ECDi :i C I}.

Let Mi := (A, Di)/EZDi, for every i C I. The set I is partially ordered and directed with respect to the relation <, where i < j iff Di C Dj. [We assume that the assignment i -4 Di is one-to-one.] For i < j we define

the mapping fij : Mi- Mj by putting: fij([a]i)

= [a]j (a c A). ([a]i is

the equivalence class of the element a relative to ECDj.) fij is a surjective homomorphism from Mi to Mj. Thus

E := (I, {Mi : i G I},{fij : i < j, i,j e I})

is a surjective directed system of Suszko-reduced models. Let D" := UiEI Di. The direct limit of the system E is isomorphic with the model M,, :=

(AD,,)/:, where 4D := U{ECDi : i c I}. By the assumption, the direct limit of the system E belongs to I(Modsu(C)). Hence, by Proposition 3.3,

?ADOO C b. Thus YAD~ C 1 = UiI ZADi and therefore EC UiE1 Di Uic SCDj. This proves the continuity of EC on FiC(A).

PROPOSITION 7.3. Let us assume that C is a finitary logic and the class I(ModSU(C)) is closed under strict homomorphic images. Then the follow ing conditions are equivalent:

(i) The operator EC is continuous on the lattice Fic(A), for all algebras A.

(ii) LS(Modsu(C)) C I(Modsu(C)). (iii) The class I(ModsU(C)) is a quasivariety.

PROOF. The equivalence of (i) and (ii) follows from Theorems 7.1 and 7.2. The proof of the implications (ii) => (iii) and (iii) #> (ii) is based on the

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The Suszko Operator. Part I. 225

following facts. Given a class K of matrices, we let Q(K) denote the small est quasivariety that includes the class K. Then Q(K) = LS(PS(K)) =

S(PR(K)). Furthermore, Q(K) is the least class of matrices which contains K and is closed with respect to Ls and Ps and, alternatively, Q(K) is the least class of matrices which contains K and is closed with respect to S and

PR, see e.g. [7, Theorem 3.2.10].

(ii) =* (iii). As Ps(I(ModSu(C) ) I(Modsu(C)) by Theorem 5.2, we have that

Q(I(ModSu(C))) = Ls(Ps(I(ModSu (C)))) = Ls(I(ModSu(C))). But (ii) says that Ls(I(ModSu(C))) = I(Modsu(C)). So Q(I(Modsu(C))) = I(ModSu(C)) which proves that I(Modsu(C)) is a quasivariety.

(iii) X (ii). This follows from the fact that every quasivariety is closed under Ls. M

PROPOSITION 7.4. If the extended Suszko operator for C has a global parame ter-free determinator E(x, y), then the class I(Modsu(C)) is closed under the formation of submatrices.

PROOF. Let M = (A, D) be a submodel of N = (B, E), where N is Suszko reduced, and suppose a -b (mod YCM) for a, b E A. Hence ,o(a, b) C

CA(D, O(a, b)) and V)(a, b) c CA(D, <(a, b)), for all pairs (, ) C E(x, y). This evidently implies that (p(a,b) c CB(E, <(a,b)) and <(a,b) c CB(E, ,o(a,b)), for all pairs (p, 0) c E(x,y). Thus a_ b (mod ZCN), which gives that a = b. O

The problem of whether the converse of the above proposition holds, that is, whether the condition S(ModSu(C)) C I(Modsu(C)) implies the existence of a global parameter-free determinator of the extended Suszko operator for C appears to be open.

We also note here that if the Suszko operator for C has a local parameter free determinator E(x, y), then Theorem 6.5 implies that E(x, y) is a deter

minator for the class Modsu(C) of proper Suszko-reduced matrices. No gen eral results implying equality of the classes Modsu(C) and I(Modsu(C)) are known as yet.

PROPOSITION 7.5. Let C be a finitary logic. If the extended Suszko operator for C has a global, finite, parameter-free determinator E(x, y), then EC is continuous on Fic(A), for every algebra A.

PROOF. Let {Di i e I} be a directed system of C-filters on an algebra A. We wish to show

EC f{Di i I} = U{ASDi i E I}.

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226 J. Czelakowski

Let D U{Di: i C I}. The inclusion "D "follows from the monotonic

ity of EC and the fact that D is a C-filter. To prove the opposite inclusion, as

sume (a, b) E ?CD for some a, b E A. Thus CA (D, Sp(a, b)) = CA (D, fb(a, b))

for every pair (o(x,y),V(x,y)) C E(x,y). But as CA is a finitary clo

sure operator on the subsets of A, the family {Di: i C I} is directed

and the set E(x, y) is finite, it follows that there exists i C I such that

CA(Di, (o(a, b)) = CA(Di, 0(a, b)) for all ((p, A) C E(x, y). This means that

(a b) e EEDi Hence (a, b) E U{EZcDj: i E I} .

It is an open question whether the converse of Proposition 7.5 holds.

THEOREM 7.6. For an arbitrary logic C the following conditions are equiv

alent:

(i) Every Suszko-reduced model M = (A, D) of C has at most one des

ignated element, i.e., Di < 1;

(ii) x y (mod ZC(x, y)), where x and y are arbitrary variables;

(iii) For every determinator E(x, y, u) of the Suszko operator for C and

for every pair (p, b) C E(x, y, u) there holds: C(x, y, p) = C(x, y, <).

(iv) For every sentential formula W of S x, y, p(x)/p(y) is a rule of C.

PROOF. The inclusion (i) =# (ii) is immediate for the matrix (S, C(x, y))

/ECC(x, y) is Suszko-reduced.

(ii) => (iii). Assume (ii) holds. Let E(x, y, u) be a determinator of

the Suszko operator for C. (ii) evidently implies that C(x, y, (x, y, u))

C(x, y, O(x, y, u)) for all pairs (ao, b) c E(x, y, u). So (iii) holds.

(iii) ?> (iv). As the set E(x, y, u) consisting of all pairs (p(x, u), p(y, a)),

where o(x, u) is an arbitrary sentential formula, is a determinator, (iii) yields

that C(x, y, (x, u)) = C(x, y, (y, u)) for all formulas ~o(x, u). From this

(iv) follows.

(iv) => (i). Assume M = (A, D) is a Suszko-reduced model of C. Let

a, b c D. By (iv), for all sentential formulas p(x, u) E S and c = cl,... ., Cn E

A,

(p(b,c) E CA(D, a, b, yc(a, c)) and (p(a, c) C CA(D, a, b, W(b, c)).

Since a, b c D, we obtain

W(b,c) e CA(D, p(a, c)) and Wo(a,c) C CA(D, (b, c)),

for all ,o and c. This means that a _ b (mod ZCD). Hence a = b because

M is Suszko-reduced.

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The Suszko Operator. Part I. 227

The above theorem is an extension of a result already proved by Suszko in 1977 (unpublished), namely that a logic C satisfies (iv) iff it satisfies the property (i'):= "Every reduced C-model has at most one designated ele ment ", see also [22]. The rules defined in (iv) are sometimes called " Suszko rules"[23]). We thus see that (i) and (i') are equivalent conditions (because they are equivalent to (iv)). We also notice the similarity of condition (ii)

with condition (ii'):= "x _ y (mod QC(x, y))", which appears in the treat ment of regularly algebraizable logics, see e.g. [7], Chapter 5. In general (ii) implies (ii') but the converse needs protoalgebraicity.

COROLLARY 7.7. Suppose C is a logic such that C(0) is non-empty and either of the conditions (i) - (iv) of Theorem 7.6 holds. Then the operator EC is injective on Fic(A), for all algebras A.

PROOF. Let D1I,1D2 E FiC(A) and D1 zA D2. Note that D1i n D2 is non empty. Assume D1 - D2 is non-empty. Let a C Di n D2, b E D1, b f D21 Then a b (mod ECD1) because ID9/ECDII = 1, and a =_ b (mod ECD2) because YiCD2 is a strict congruence on (A, D2). Thus YjD1 7& ZCD2. c

In the general case, one may ask for conditions equivalent to the injec tivity of EC on filter lattices in arbitrary algebras A. We note the following remark:

THEOREM 7.8. For any logic (S, C) the following conditions are equivalent:

(i) SAC is injective on Fic(A), for all algebras similar to S.

(ii) For any algebra and any filter D E FiC(A), the quotient set D/EjD {[a] : a E D} is the least C-flter in the algebra A/EYD. ([a] is

the equivalence class of a with respect to ECD).

(iii) For every Suszko-reduced matrix (A, D) C I(Modsu(C)), D is the least C-filter in the algebra A.

PROOF. (i) == (ii). Assume (i) holds. Fix an algebra A and let D E Fic(A). Furthermore assume F0 is the least C-filter in the quotient algebra B :=

A/ECD. We claim Fo = D/EZCD. As D/YECD is a C-filter in B, we ob viously have that F0 C DE/ZCD. Since the matrix (A/ECD,D/ZECD) is Suszko-reduced, the congruence EZC(D/ECD) is the identity relation on B. Hence, by the monotonicity of EC, IdB C EYFO C ECZ(D/EYCD) = IdB. Hence ECFO = ?C(D/EAD) which gives that F0 = D/ECD by the injec tivity of EC on Fic(B). So D/ECD is the smallest C-filter on A/ECD.

(ii) ?= (iii). This is immediate.

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228 J. Czelakowski

(iii) =X (ii). Assume (iii). Let D E Fic(A), where A is an arbitrary algebra similar to S. As the matrix (A/ECD, D/IED) is Suszko-reduced,

D/ECD is the least C-filter in Fic(A/ECD) by (iii). So (ii) holds. (ii) =* (i). Assume (ii). Let A be an algebra and let D and D' be C

filters in A such that SAD - SCD'. The Suszko-reduced matrices (A/ECD, D/ECD) and (A/ECD', D'/ECD') thus have the same underlying algebra. By (ii), D/IED is the least C-filter in A/ECD and D'/ECD' is the least C-filter in A/ECD'. Since the algebras A/ECD and A/ECD' are identical, D/ECD = D/ECD'. This and ECD = ECD' give D = D'.

If C is protoalgebraic, the above theorem gives a known property of Q and Leibniz reductions, see e.g. [9] or [7, Theorem 1.6.7].

EXAMPLE 7.9. Let S be the language which involves only two binary con nectives A, V and one constant 1 (Veritas). (Adjoining the constant 1 to the vocabulary of S has a purely technical meaning. It simplifies the formula tion of the main result.) Let C be the {/A, V, 1}-reduct of classical sentential logic. We have:

(*) I(ModSu(C)) coincides with the class of all matrices M = (A, { 1 }), where A is a distributive lattice with unit and 1 is the unit element of A.

I(Modsu(C)) can be thus identified with the class D1 of distributive lattices with unit.

(*) follows from Theorem 5.6 and the fact that Alg(C) coincides with the class of distributive lattices (with unit), see [15, Section 5.1.1].

The above observations close the first part of the paper. The second part, which will appear, will be concerned with the structure of the class of proper Suszko-reduced matrices for various deductive systems. It seems that the class Modsu(C) forms a right algebraic semantics for a non-protoalgebraic logic C. It should be underlined that the class Modsu(C) is defined by

means of genuinely logical procedures, viz. by taking natural extensions of C and then factoring the resulting Lindenbaum matrices by their Suszko congruences.

In the second part also some weak deduction theorems for deductive systems will be investigated. They are called equationally assisted deduction theorems. Each logic can be proved to admit such a theorem.

Acknowledgments

I would like to express my gratitude to Josep Maria Font, Ramon Jansana and Don Pigozzi who read critically the first version of the paper. They

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The Suszko Operator. Part I. 229

checked through the whole text and brought a number of inaccuracies to my attention. They made numerous useful suggestions which considerably improved the final draft and simplified proofs of many theorems presented in the paper.

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JANUSZ CZELAKOWSKI Institute of Mathematics and Computer Science

Opole University Oleska 48 45-052 Opole, Poland j cze1Qmath .uni .opole .p1

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