Blok, W. J. and Pigozzi, D., Algebraizable Logics, Memoirs of the A.M.S. 77 Nr. 396, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....22, Santiago, CHILE Marta Sagastume Departamento de Matematicas Universidad Nacional de La Plata La Plata, ARGENTINA Abstract In [6] C. C. Chang introduced a natural generalization of Lukasiewicz infinite valued propositional logic L. In this logic the truth values are extended from the interval [0,1] to the interval [ 1,1] We will call L # the logic whose designated values are those greater or equal than 0. Chang calls this logic p # [0] In this semantics, for a truth assignment v the value of the negation is v(#) v(#) This implies that there are sentences for which v(#) v(#) 0 ....

....Sagastume Departamento de Matematicas Universidad Nacional de La Plata La Plata, ARGENTINA Abstract In [6] C. C. Chang introduced a natural generalization of Lukasiewicz infinite valued propositional logic L. In this logic the truth values are extended from the interval [0,1] to the interval [ 1,1]. We will call L # the logic whose designated values are those greater or equal than 0. Chang calls this logic p # [0] In this semantics, for a truth assignment v the value of the negation is v(#) v(#) This implies that there are sentences for which v(#) v(#) 0 , that is, both ....

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Blok, W. J. and Pigozzi, D., Algebraizable Logics, Memoirs of the A.M.S., 77 Nr. 396, 1989.

....the novel notion of cryptofibring . Other interesting lines of research require a deep understanding of the process of algebraization of logics, putting in context the notion of fullness and the role that it plays in the completeness results, bringing us closer to the rich field of algebraic logic [4, 1]. We are also interested in studying the representation of fibring in logical frameworks, by capitalizing on the theory of general logics [15] Finally, future work must also cover transfer results for other relevant properties, like decidability, complexity or interpolation. ....

W. Blok and D. Pigozzi. Algebraizable Logics, volume 77. Memoires AMS, 1989.

....sense and that the logical operators give rise to normal operators in the Lindenbaum algebra. For examples of logical systems that are not algebraizable in the above sense and for a more general investigation of the question of algebraizability of a logic we refer the reader to Blok and Pigozzi [7]. For lack of space we do not give a thorough review, nor do we present the proof systems associated to the various types of algebras. For details we refer the reader to Ono [36, 35] Note that the rule of association is maintained. 28 operators ffi (times) and (reverse implication) and ....

W. J. BLOK and D. PIGOZZI (1989), Algebraizable Logics, Memoirs of the American Mathematical Society 77, number 396, Providence, Rhode Island, USA.

....that algebraic logic (AL for short) is a broad and very active eld, and a handbook should cover at least the most important branches. The papers collected so far do not cover all these branches. Two of these branches which we particularly miss are relation algebras and general AL in the sense of [4] and [3] On the other hand, the papers collected so far do cover many important branches and they, when put together, satisfy the above indicated need to a considerable extent. Therefore we think that we can serve the scienti c community best if we publish the already collected papers in a ....

.... algebras, Pinter s substitution cylindric algebras, and many others) The universal algebraic or general part contains e.g. Boolean algebras with operators and their duality theory via Kripke style models of polymodal logics, general (or abstract) AL in the sense of the Blok and Pigozzi school [4], and general AL with model theoretic semantics as represented by, e.g. 3] A uni ed development of the latter two approaches is in preparation ( 1] A common feature of the theories of the core part of AL is that they study algebras whose elements are relations of ranks 2. The quest for ....

W. J. Blok and D. L. Pigozzi. Algebraizable Logics. Memoirs Amer. Math. Soc., 77, 396:vi+78, 1989.

....of Funding for this paper has been provided by FONDECYT grant 1930551. Additional funding was provided by a grant by the Fundaci on Andes, Programa C12600 4 Segunda Etapa 1994, for a visit to Iowa State University. 1 2 R. A. LEWIN, I. F. MIKENBERG, AND M. G. SCHWARZE Blok and Pigozzi, [2] and [3] The sistems SP are not complete with respect to the semantics given in that paper. In this paper we propose a matrix semantics for SAL , structural annotated logic based on , which are extensions of SP . Since any extension of an algebraizable system is itself algebraizable, for ....

....2 . 3. Matrix Semantics for SAL In this section we de ne a family EM of very simple matrices with respect to which the system SAL is sound. Based on these, we build a family M of more complex matrices that is a matrix semantics for system SAL . We rst give a few basic de nitions. See [2], 3] or [5] for details. De nition 3.1. A valuation into the matrix M = hA; Di is the unique homomorphism that extends a function u : P A 8 R. A. LEWIN, I. F. MIKENBERG, AND M. G. SCHWARZE to u : F A: We often call u a valuation since it totally determines the homomorphism. The ....

Blok, W. J. and Pigozzi, D., Algebraizable Logics, Memoirs of the A.M.S., 77 Nr. 396 (1989).

....that algebraic logic (AL for short) is a broad and very active field, and a handbook should cover at least the most important branches. The papers collected so far do not cover all these branches. Two of these branches which we particularly miss are relation algebras and general AL in the sense of [4] and [3] On the other hand, the papers collected so far do cover many important branches and they, when put together, satisfy the above indicated need to a considerable extent. Therefore we think that we can serve the scientific community best if we publish the already collected papers in a ....

.... algebras, Pinter s substitution cylindric algebras, and many others) The universal algebraic or general part contains e.g. Boolean algebras with operators and their duality theory via Kripke style models of polymodal logics, general (or abstract) AL in the sense of the Blok and Pigozzi school [4], and general AL with model theoretic semantics as represented by, e.g. 3] A unified development of the latter two approaches is in preparation ( 1] A common feature of the theories of the core part of AL is that they study algebras whose elements are relations of ranks # 2. The quest for ....

W. J. Blok and D. L. Pigozzi. Algebraizable Logics. Memoirs Amer. Math. Soc., 77, 396:vi+78, 1989.

....system P1 introduced by Sette in [6] is algebraizable. In this paper we study the main features of the class of algebras thus obtained. The main results are a complete description of the free algebras in n generators and that this is not a congruence modular quasi variety. Introduction In [1] Blok Pigozzi develop a theory of algebraizability of deductive systems that generalizes the usual Lindenbaum Tarski process. Besides giving criteria to determine if a system is algebraizable or not, they provide means to obtain the algebraic counterpart of the system. In [3] the authors used ....

....[2] are not algebraizable. This fact was proved by Mortensen in [5] for a shorter proof using Blok Pigozzi s theory see [4] In this paper we will give a characterization of these algebras, we will describe its free algebras and study some aspects of its lattices of congruences. Following [1] theorem 2.17 and [6] the axioms that de ne P1 algebras are equivalent to the following. De nition 1. A P1 algebra is an algebra A = hA; 0 ; 1i where is a binary operation, 0 is a unary operation and 1 is a constant that verify the identities: A1. x (y x) 1, A2. x (y z) ....

Blok, W.J. and Pigozzi, D. Algebraizable Logics, Memoirs AMS, 77, N 396 (1989).

....Free Annotated Algebras Renato A. Lewin, Irene F. Mikenberg and Mar a G. Schwarze Ponti cia Universidad Cat olica de Chile Santiago Chile Abstract In [4] the authors proved that certain systems of annotated logics are algebraizable in the sense of [1]. Later in [5, 6] the study of the associated quasi varieties of annotated algebras is initiated. In this paper we continue the study of the these classes of algebras, in particular, we report some recent results about the free annotated algebras. 1 Preliminaries For the sake of conciseness, we ....

....the study of the these classes of algebras, in particular, we report some recent results about the free annotated algebras. 1 Preliminaries For the sake of conciseness, we will assume that the reader is familiar with the general theory of algebraizability of deductive systems as developed in [1]. This paper is one in a series in which we have applied the theory to a family of deductive systems known as annotated logics We state here without proof the main results about them obtained in [4, 5, 6] 1.1 Annotated Logics Annotated logics were introduced in [7] by V. S. Subrahmanian as ....

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Blok, W. J. and Pigozzi, D., Algebraizable Logics, Memoirs of the A.M.S. 77 Nr. 396 (1989).

....SP such that a formula is provable in P if and only if it is provable in SP using some additional premises. For details, see [11] Finally we studied the algebraizability of these systems using as main framework the theory of algebraization of deductive systems developed by Blok and Pigozzi in [3]. The main result proven in [11] is that these systems are algebraizable. Moreover, we proved that an annotated logic is nitely algebraizable (i.e. it has a nite system of congruence formulas) if and only if the lattice of annotation constants is nite. All axioms and inference rules of SP are ....

....the usual submatrix, direct product and ultraproduct matrix theoretic operators. This theorem states that M is the smallest matrix quasivariety containing K. 4. 1 The Leibniz Operator One of the main tools in algebraic logic is the so called Leibniz operator, which is extensively studied in [3, 5, 9, 10], and other places. We give here the main de nitions and properties for future reference. For any algebra A and D A, we de ne the Leibniz congruence relation on A over D A (D) fha; bi : A (a; c 1 ; c n ) 2 D i A (b; c 1 ; c n ) 2 D; for any formula (x; y 1 ; ....

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Blok, W. J. and Pigozzi, D., Algebraizable Logics, Memoirs of the A.M.S., Nr. 396, vol. 77, 1989.

....used, an application of which was the main motivation for this paper, are best suited to work syntactically. Finally, in Section 4, we study the algebraizability of these systems using as main framework the theory of algebraization of deductive systems developed by Blok and Pigozzi in [3]. The main result proven here, is that these systems are, using the terminology of [4] weakly congruential, namely, they have an in nite system of congruence 1formulas. Moreover, we characterize the class of all annotated logics that are algebraizable, i.e. that have a nite system of congruence ....

....annotated logics that are algebraizable, i.e. that have a nite system of congruence formulas. One should note that strictly speaking, Blok Pigozzi s theory does not apply to these systems since they contain an in nitary rule of inference and thus, there are in nite proofs, in the terminology of [3], they are not standard. Nevertheless, the methods developed there can be extended to arbitrary systems through the intrinsic characterization of algebraizable deductive systems via the Leibniz equality function. See Theorem 4.2 of [3] It should be noted that Herrman in [8] does this ....

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Blok, W. J. and Pigozzi, D., Algebraizable Logics, Memoirs of the A.M.S., 77 Nr. 396 (1989).

....ole in some parts of the paper. To avoid misunderstandings, and following recent practice, we use algebraizable for CZELAKOWSKI s [11] and HERRMANN s [26] notion (also called possibly infinitely algebraizable in [27,28] and use finitely algebraizable for the original notion by BLOK and PIGOZZI [2] but applied also to non finitary logics as is done in [11,15,26] Moreover, we consider the class of weakly algebraizable logics, treated extensively in [12,15] All these kinds of logics can be defined by properties of the so called Leibniz operator introduced in [1] i.e. the mapping #A ....

....it is at the same time equivalential (resp. finitely equivalential) and weakly algebraizable. See [15] for details. ## Notice that in case S is both finitary and finitely equivalential then the finitely algebraizable in Proposition 21 amounts to being algebraizable in the original sense of [2]. Another observation is that there is no point in considering what happens if S is placed higher in the Abstract Algebraic Logic hierarchy: If it is weakly algebraizable (a fortiori, if it is algebraizable) then #A is injective by definition and thus every S filter is Leibniz; therefore S ....

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BLOK, W. J., AND PIGOZZI, D. Algebraizable logics, vol. 396 of Mem. Amer. Math. Soc. A.M.S., Providence, January 1989.

....completeness. One consequence of this is an alternative proof of Theorem 4; the argument is parallel to the one we used after Theorem 4 to supply a proof of Theorem 1, hence we omit it. The next result classifies # # L according to recent criteria of abstract algebraic logic, as introduced in [1] and further developed in [11] and [12] Theorem 7. The logic # # L is finitely, strongly and regularly algebraizable and its equivalent algebraic semantics is the class of discrete epistemic algebras. Proof. # # L is an expansion of CPC, a logic that is finitely algebraizable and has the ....

....of CPC, a logic that is finitely algebraizable and has the class of all Boolean algebras as its equivalent algebraic semantics. The equivalence formulas are p # q , q # p and the defining equation is p # #. By the syntactic characterization of algebraizability given in Theorem 4. 7 of [1], in order to check that # # L is also finitely algebraizable with the same equivalence formulas and defining equation, it is enough to 9 check that the property of congruentiality also holds for the extra language; in the present case we should show that p # q , q # p # # L #p # ....

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Blok, W. J., and Pigozzi, D. Algebraizable logics, vol. 396 of Mem. Amer. Math. Soc. A.M.S., Providence, January 1989.

....logic. I give a systematic presentation of the method of diagrams for first order languages without equality. Keywords: equality free logic, model theory 1 Introduction The interest for the study of languages without equality has its origin in the works of W. Blok and D. Pigozzi, 1] 2] and [3]. Equality free Logic can be seen as a bridge between two disciplines: Model Theory and Algebraic Logic. It was first observed by S. L. Bloom in [4] that to any propositional deductive system we can associate an equality free strict universal Horn theory. For this reason, in order to study the ....

....concepts which lay in the background of both disciplines and that allow the development of this study, the notion of Leibniz congruence and the notion of relative relation. W. Blok and D. Pigozzi introduced the concept of relative relation for the special case of logical matrices in [1] and in [3] they made an extensive use of what they named the Leibniz congruence. Motivated by their works a general classical model theoretical study of this logic was carried on in [5] 8] 9] 10] and [11] In [5] we developed back and forth methods for equality free languages and di#erent ....

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W. J. Blok, D. Pigozzi (1989): Algebraizable Logics, Memoirs of the Am. Math. Soc. 396.

.... of set theory without variables has been shown to provide adequate foundations for set theory and arithmetic (Tarski and Givant, 1987) and has been used to model description logics (Brink and Schmidt, 1992) A number of precise characterizations of algebraizable logics have been developed (Blok and Pigozzi, 1989). In the mid 1980s At Kaci (1984, 1986) gave a lattice theoretic model of knowledge base languages with operational semantics through term rewriting that resolved many of the issues of complexity and deduction algorithms for term subsumption knowledge representation systems. This y calculus is ....

Blok, W.J. and Pigozzi, D. (1989). Algebraizable Logics. Providence, Rhode Island, American Mathematical Society.

.... to (are equivalent with) representation theorems in algebra, cf. e.g. AKNS 94] 2 first step in algebraic logic is that, to any logic L, one associates a class Alg(L) of algebras (this can be done under relatively weak conditions on L, see e.g. Andr eka et al. AKNS 94] Blok Pigozzi [BP 89] BP] Henkin Monk Tarski [HMT 85] Chapter 4.3) For example, if L is a multi modal logic then Alg(L) is a class of BAO s (a class of similar algebras having a Boolean reduct; the extra Boolean operations correspond to the modalities of L) Then one builds a kind of bridge between logic (where ....

.... lives ) by, e.g. proving theorems of the following kind: Logic L has property P iff the class Alg(L) of algebras has property alg(P ) where alg(P ) is a natural algebraic property. More on this subject can be found, e.g. in the Introduction of Andr eka et al. AKNS 94] and in BlokPigozzi [BP 89] BP] In Sections 4 6 we prove (or state) equivalence theorems of the above kind. Namely, in [Mak 91] and [Mak 79] Maksimova proved that a normal modal logic (with one unary modality) has the Craig interpolation property iff the corresponding class of algebras has the superamalgamation ....

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W. J. Blok, D. Pigozzi, Algebraizable logics, Memoirs Amer. Math. Soc., Vol. 77, No. 396, (1989), vi+78 pp.

....the same interpretation. Yet we say that Sigma defines implicitly because we did not explicitly write down a formula without 1 Basically, in this area one can distinguish two different approaches. First there is the work of (among others) W. Blok and D. Pigozzi, in this paper represented by [BP 89] Second, there is the so called Budapest School, lead by H. Andr eka, I. N emeti and I. Sain, cf. e.g. NA 94] AKNS 94] but also [HMT 85, x5.6] In this paper we follow the second tradition, but we have tried to write this paper in such a way as to make it understandable for readers more ....

....all the notions introduced below, will be explained in full detail in Section 2. For the time being, we only mention the following. By a strongly nice logic we mean the same as in [AKNS 94] This is basically the same as the notion of an algebraizable logic in the sense of Blok Pigozzi, cf. e.g. BP 89] Furthermore, by Alg j= L) we mean a specific class of algebras corresponding to the logic L in the way described above. Intuitively, one can think of Alg j= L) as the class of Lindenbaum Tarski algebras of L. Finally, the patchwork property of models can be thought of as an amalgamation ....

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W.J. Blok, D. Pigozzi, Algebraizable logics, Memoirs Amer. Math. Soc. Vol. 77, No. 396 (1989), vi+78 p.

....first order predicate logic without equality. In considering isomorphisms between models of logic without equality, it is natural to identify elements which are indistinguishable from each other. That is, it is natural to restrict attention to models which are reduced in the following sense (see [BP], CDJ] D] DJ] Definition 2.1 A model M for L is said to be reduced if for any pair of elements a; b 2 M , we have a = b if and only if for every formula (x; y) of L, M j= 8 u[ a; u) b; u) 1) In general, two elements a; b 2 M are said to be Leibniz congruent, in symbols a j b, ....

W.J.Blok and D. Pigozzi, Algebraizable logics, Memoirs of the A.M.S. no. 396 (1989).

....algebra Fm Pi modulo renaming bound variables. For the proof theoretic FOL, is determined by a system I of ordinary inference rules of the predicate calculus, which must be written without any reservations about occurrence of variables in quantified formulas, this is indeed possible (see, eg, [3]) Thus, jj Gamma (I) For the model theoretic FOL, is determined by means of models: Gamma A Pi OE iff Gamma j= Mod Pi) Fm Pi OE. The right hand side relation can be presented by validity in matrices, j= Mod Pi) Fm Pi = j= M) A Pi , in a usual way: M = f(MX ; D) j MX = ....

W. Blok and D. Pigozzi. Algebraizable logics. Memoirs Amer.Math.Soc., 77, 1989.

....sense and that the logical operators give rise to normal operators in the Lindenbaum algebra. For examples of logical systems that are not algebraizable in the above sense and for a more general investigation of the question of algebraizability of a logic we refer the reader to Blok and Pigozzi [7]. 18 For lack of space we do not give a thorough review, nor do we present the proof systems associated to the various types of algebras. For details we refer the reader to Ono [36, 35] 19 Note that the rule of association is maintained. operators ffi (times) and (reverse implication) and ....

W. J. BLOK and D. PIGOZZI (1989), Algebraizable Logics, Memoirs of the American Mathematical Society 77, number 396, Providence, Rhode Island, USA.

.... work [22, 23] In the context of logic the basic idea is an old one, going back to the beginnings of algebraic logic and the Lindenbaum Tarski process [38] More recently it has been the principal feature of abstract algebraic logic (which can also be viewed as coalgebraic logic; see for instance [4, 8, 30]) In computer science it has also been around for a long time; see [22] for a systematic development of the non proof theoretical part of the theory in the context of correct subtyping. A more extensive development that includes parts of the proof theory but not subtyping can be found in the ....

....congruence satisfying a certain property can be traced back to the process of forming the LindenbaumTarski algebra of the classical propositional calculus by Tarski [38] A logical analog of Theorem 7. 3 can be found in [36] Subsequently these ideas formed the basis of abstract algebraic logic [4, 8, 12, 30] The dichotomy between behavioral equivalence as a relation between objects and between environments seems to be novel, although elements of the latter (in the form of homomorphic generalized relations) can be found in [37] in the context of type theory. As mentioned above Theorem 7.3 is a ....

Willem Blok and Don Pigozzi, Algebraizable Logics. Mem. Amer. Math. Soc., vol. 396, Amer. Math. Soc., Providence, 1989.

....is a necessary condition in all cases. We get around the problem in the case of PL by constructing a (logical) matrix M(L) whose associated logic PM(L) which is necessarily structural, naturally simulates PL . We then show that, if L is nite, P M(L) is nitely algebraizable (in the sense of [4]) and its equivalent algebraic semantics is the quasivariety generated by a special quotient M (L) of the underlying algebra of the matrix M(L) The simulation of PL by PM(L) is then composed with the bisimulation between PM(L) and the equational logic of M (L) that arises naturally from ....

....with previous work and outline of paper. Lewin, Mikenberg, and Schwarze [10] introduce for each PL a new, structural, annotated system SPL that is closely related to PL in the sense that it simulates its deductive process in a natural way. They prove SPL is nitely algebraizable in the sense of [4] just in case the lattice L is nite. In [11, 12] they consider another nitely algebraizable annotated logical system SALL , called structural annotated logic based on L, that is an axiomatic extension of SPL . They investigate its equivalent algebraic semantics and a matrix semantics for SALL . ....

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Blok, W. J. and Pigozzi, D., Algebraizable Logics, Memoirs of the A.M.S., 77 Nr. 396, 1989.

....get around the problem in the case AN ANNOTATED LOGIC DEFINED BY A MATRIX 3 of PL by constructing a (logical) matrix M(L) whose associated logic PM(L) which is necessarily structural, naturally simulates PL . We then show that, if L is finite, PM(L) is finitely algebraizable (in the sense of [4]) and its equivalent algebraic semantics is the quasivariety generated by a special quotient M (L) of the underlying algebra of the matrix M(L) The simulation of PL by P M(L) is then composed with the bisimulation between PM(L) and the equational logic of M (L) that arises naturally from ....

....with previous work and outline of paper. Lewin, Mikenberg, and Schwarze [10] introduce for each PL a new, structural, annotated system SPL that is closely related to PL in the sense that it simulates its deductive process in a natural way. They prove SPL is finitely algebraizable in the sense of [4] just in case the lattice L is finite. In [11, 12] they consider another finitely algebraizable annotated logical system SALL , called structural annotated logic based on L, that is an axiomatic extension of SPL . They investigate its equivalent algebraic semantics and a matrix semantics for SALL ....

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Blok, W. J. and Pigozzi, D., Algebraizable Logics, Memoirs of the A.M.S., 77 Nr. 396, 1989.

....to (M1) M7) pocrims also satisfy the following properties: M8) x x 1, M9) x (y z) y (x z) M10) If x y then y z x z and z x z y, M11) x (x y) y, M12) x y x. The quasivariety M of all pocrims is the equivalent algebraic semantics in the sense of [5] of the algebraizable deductive system S M below, as shown by J. Raftery and J. Van Alten [35] The axioms of S M are: B) p q) r p) r q) C) p (q r) q (p r) K) p (q p) OK1) p (q (p Delta q) OK2) p (q r) p Delta q) ....

W.J. Blok and D. Pigozzi, Algebraizable Logics, Mem. Amer. Math. Soc. 396 (1989).

....classes of algebras in which congruences can be represented by suitable subsets in a uniform and definable way, and to characterize these subsets by appropriate closure properties, we adopt a logical perspective, possibly hinted at earlier by A. Robinson in [Rob] and at the core of the monograph [BP89]. We view the closure systems of (relative) 1991 Mathematics Subject Classification. Primary: 08C15, 03B22, 03G25. Secondary: 06D99. 1 2 W.J. BLOK AND J.G. RAFTERY congruences of algebras in a quasivariety as models of a logic of a certain kind. The requirement that congruences be representable ....

....of the relative congruences by suitable subsets, then these subsets are just the strong ideals, and hence a syntactic characterization of class is available. This situation obtains precisely when is an equivalent algebraic semantics of the deductive system S with defining equations , and [BP89] provides the tools to decide when this is so. In particular, S is always algebraizable in this case. We illustrate the theory by showing that the variety of Sugihara algebras referred to above, although not point regular, is regular with respect to a certain translation . We conclude ....

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W.J. Blok, D. Pigozzi, "Algebraizable Logics", Memoirs of the American Mathematical Society, Number 396, Amer. Math. Soc., Providence, 1989.

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Blok, W. J. and Pigozzi, D., Algebraizable Logics, Memoirs of the A.M.S. 77 Nr. 396, 1989.

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