12 research outputs found

    A note on drastic product logic

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    The drastic product D*_D is known to be the smallest tt-norm, since xDy=0x *_D y = 0 whenever x,y<1x, y < 1. This tt-norm is not left-continuous, and hence it does not admit a residuum. So, there are no drastic product tt-norm based many-valued logics, in the sense of [EG01]. However, if we renounce standard completeness, we can study the logic whose semantics is provided by those MTL chains whose monoidal operation is the drastic product. This logic is called S3MTL{\rm S}_{3}{\rm MTL} in [NOG06]. In this note we justify the study of this logic, which we rechristen DP (for drastic product), by means of some interesting properties relating DP and its algebraic semantics to a weakened law of excluded middle, to the Δ\Delta projection operator and to discriminator varieties. We shall show that the category of finite DP-algebras is dually equivalent to a category whose objects are multisets of finite chains. This duality allows us to classify all axiomatic extensions of DP, and to compute the free finitely generated DP-algebras.Comment: 11 pages, 3 figure

    Inconsistency lemmas in algebraic logic

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    In this paper, the inconsistency lemmas of intuitionistic and classical propositional logic are formulated abstractly. We prove that, when a (finitary) deductive system is algebraized by a variety K, then has an inconsistency lemma—in the abstract sense—iff every algebra in K has a dually pseudo-complemented join semilattice of compact congruences. In this case, the following are shown to be equivalent: (1) has a classical inconsistency lemma; (2) has a greatest compact theory and K is filtral, i.e., semisimple with EDPC; (3) the compact congruences of any algebra in K form a Boolean lattice; (4) the compact congruences of any A ∈ K constitute a Boolean sublattice of the full congruence lattice of A. These results extend to quasivarieties and relative congruences. Except for (2), they extend even to protoalgebraic logics, with deductive filters in the role of congruences. A protoalgebraic system with a classical inconsistency lemma always has a deduction-detachment theorem (DDT), while a system with a DDT and a greatest compact theory has an inconsistency lemma. The converses are false.http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1521-3870hb201

    The General Apple Property and Boolean terms in Integral Bounded Residuated Lattice-ordered Commutative Monoids

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    In this paper we give equational presentations of the varieties of {\em integral bounded residuated lattice-ordered commutative monoids} (bounded residuated lattices for short) satisfying the \emph{General Apple Property} (GAP), that is, varieties in which all of its directly indecomposable members are local. This characterization is given by means of Boolean terms: \emph{A variety V\mathsf{V} of \brl s has GAP iff there is an unary term b(x)b(x) such that V\mathsf{V} satisfies the equations b(x)¬b(x)b(x)\lor\neg b(x)\approx \top and (xkb(x))(b(x)k.x)(x^k\to b(x))\cdot(b(x)\to k.x)\approx \top, for some k>0k>0}. Using this characterization, we show that for any variety V\mathsf{V} of bounded residuated lattice satisfying GAP there is k>0k>0 such that the equation k.xk.¬xk.x\lor k.\neg x\approx \top holds in V\mathsf{V}, that is, VWLk\mathsf{V} \subseteq \mathsf{WL_\mathsf{k}}. As a consequence we improve Theorem 5.7 of \cite{CT12}, showing in theorem that a\emph{ variety of \brls\ has Boolean retraction term if and only if there is k>0k>0 such that it satisfies the equation k.xkk.(¬k)nk.x^k\lor k.(\neg k)^n\approx\top.} We also see that in Bounded residuated lattices GAP is equivalent to Boolean lifting property (BLP) and so, it is equivalent to quasi-local property (in the sense of \cite{GLM12}). Finally, we prove that a variety of \brl s has GAP and its semisimple members form a variety if and only if there exists an unary term which is simultaneously Boolean and radical for this variety.Comment: 25 pages, 1 figure, 2 table

    On structural completeness versus almost structural completeness problem : a discriminator varieties case study

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    We study the following problem: determine which almost structurally complete quasivarieties are structurally complete. We propose a general solution to this problem and then a solution in the semisimple case. As a consequence, we obtain a characterization of structurally complete discriminator varieties. An interesting corollary in logic follows: Let L be a propositional logic/deductive system in the language with formulas for verum, which is a theorem, and falsum, which is not a theorem. Assume also that L has an adequate semantics given by a discriminator variety. Then L is structurally complete if and only if it is maximal. All such logics/deductive systems are almost structurally complete.submittedVersionFil: Campercholi, Miguel Alejandro Carlos. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.Fil: Stronkowski, Michal M. Warsaw University of Technology. Faculty of Mathematics and Information Sciences; Polonia.Fil: Vaggione, Diego José. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.Matemática Pur

    Relative congruence formulas and decompositions in quasivarieties

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    Quasivarietal analogues of uniform congruence schemes are discussed, and their relationship with the equational definability of principal relative congruences (EDPRC) is established, along with their significance for relative congruences on subalgebras of products. Generalizing the situation in varieties, we prove that a quasivariety is relatively ideal iff it has EDPRC; it is relatively filtral iff it is relatively semisimple with EDPRC. As an application, it is shown that a finitary sentential logic, algebraized by a quasivariety K, has a classical inconsistency lemma if and only if K is relatively filtral and the subalgebras of its nontrivial members are nontrivial. A concrete instance of this result is exhibited, in which K is not a variety. Finally, for quasivarieties M⊆K, we supply some conditions under which M is the restriction to K of a variety, assuming that K has EDPRC.The second author was supported in part by the National Research Foundation of South Africa (UID 85407).https://link.springer.com/journal/122482018-11-27hj2017Mathematics and Applied Mathematic

    Structural and universal completeness in algebra and logic

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    In this work we study the notions of structural and universal completeness both from the algebraic and logical point of view. In particular, we provide new algebraic characterizations of quasivarieties that are actively and passively universally complete, and passively structurally complete. We apply these general results to varieties of bounded lattices and to quasivarieties related to substructural logics. In particular we show that a substructural logic satisfying weakening is passively structurally complete if and only if every classical contradiction is explosive in it. Moreover, we fully characterize the passively structurally complete varieties of MTL-algebras, i.e., bounded commutative integral residuated lattices generated by chains.Comment: This is a preprin

    New perspectives on semi-primal varieties

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    peer reviewedWe study varieties generated by semi-primal lattice-expansions by means of category theory. We provide a new proof of the Keimel-Werner topological duality for such varieties and, using similar methods, establish its discrete version. We describe multiple adjunctions between the variety of Boolean algebras and the variety generated by a semi-primal lattice-expansion, both on the topological side and explicitly algebraic. In particular, we show that the Boolean skeleton functor has two adjoints, both defined by taking certain Boolean powers, and we identify properties of these adjunctions which fully characterize semi-primality of an algebra. Lastly, we give a new characterization of canonical extensions of algebras in semi-primal varieties in terms of their Boolean skeletons

    Many-valued coalgebraic logic over semi-primal varieties

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    We study many-valued coalgebraic logics with semi-primal algebras of truth-degrees. We provide a systematic way to lift endofunctors defined on the variety of Boolean algebras to endofunctors on the variety generated by a semi-primal algebra. We show that this can be extended to a technique to lift classical coalgebraic logics to many-valued ones, and that (one-step) completeness and expressivity are preserved under this lifting. For specific classes of endofunctors, we also describe how to obtain an axiomatization of the lifted many-valued logic directly from an axiomatization of the original classical one. In particular, we apply all of these techniques to classical modal logic
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