16 found
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  1.  42
    Distinguished algebraic semantics for t -norm based fuzzy logics: Methods and algebraic equivalencies.Petr Cintula, Francesc Esteva, Joan Gispert, Lluís Godo, Franco Montagna & Carles Noguera - 2009 - Annals of Pure and Applied Logic 160 (1):53-81.
    This paper is a contribution to Mathematical fuzzy logic, in particular to the algebraic study of t-norm based fuzzy logics. In the general framework of propositional core and Δ-core fuzzy logics we consider three properties of completeness with respect to any semantics of linearly ordered algebras. Useful algebraic characterizations of these completeness properties are obtained and their relations are studied. Moreover, we concentrate on five kinds of distinguished semantics for these logics–namely the class of algebras defined over the real unit (...)
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  2.  43
    On the standard and rational completeness of some axiomatic extensions of the monoidal t-Norm logic.Francesc Esteva, Joan Gispert, Lluís Godo & Franco Montagna - 2002 - Studia Logica 71 (2):199 - 226.
    The monoidal t-norm based logic MTL is obtained from Hájek''s Basic Fuzzy logic BL by dropping the divisibility condition for the strong (or monoidal) conjunction. Recently, Jenei and Montgana have shown MTL to be standard complete, i.e. complete with respect to the class of residuated lattices in the real unit interval [0,1] defined by left-continuous t-norms and their residua. Its corresponding algebraic semantics is given by pre-linear residuated lattices. In this paper we address the issue of standard and rational completeness (...)
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  3.  16
    On the Standard and Rational Completeness of some Axiomatic Extensions of the Monoidal T-norm Logic.Francesc Esteva, Joan Gispert, Lluís Godo & Franco Montagna - 2002 - Studia Logica 71 (2):199-226.
    The monoidal t-norm based logic MTL is obtained from Hájek's Basic Fuzzy logic BL by dropping the divisibility condition for the strong (or monoidal) conjunction. Recently, Jenei and Montgana have shown MTL to be standard complete, i.e. complete with respect to the class of residuated lattices in the real unit interval [0,1] defined by left-continuous t-norms and their residua. Its corresponding algebraic semantics is given by pre-linear residuated lattices. In this paper we address the issue of standard and rational completeness (...)
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  4.  43
    Perfect and bipartite IMTL-algebras and disconnected rotations of prelinear semihoops.Carles Noguera, Francesc Esteva & Joan Gispert - 2005 - Archive for Mathematical Logic 44 (7):869-886.
    IMTL logic was introduced in [12] as a generalization of the infinitely-valued logic of Lukasiewicz, and in [11] it was proved to be the logic of left-continuous t-norms with an involutive negation and their residua. The structure of such t-norms is still not known. Nevertheless, Jenei introduced in [20] a new way to obtain rotation-invariant semigroups and, in particular, IMTL-algebras and left-continuous t-norm with an involutive negation, by means of the disconnected rotation method. In order to give an algebraic interpretation (...)
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  5.  8
    On Some Varieties of MTL-algebras.Carles Noguera, Francesc Esteva & Joan Gispert - 2005 - Logic Journal of the IGPL 13 (4):443-466.
    The study of perfect, local and bipartite IMTL-algebras presented in [29] is generalized in this paper to the general non-involutive case, i.e. to MTL-algebras. To this end we describe the radical of MTL-algebras and characterize perfect MTL-algebras as those for which the quotient by the radical is isomorphic to the two-element Boolean algebra, and a special class of bipartite MTL-algebras,.
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  6.  26
    On triangular norm based axiomatic extensions of the weak nilpotent minimum logic.Carles Noguera, Francesc Esteva & Joan Gispert - 2008 - Mathematical Logic Quarterly 54 (4):387-409.
    In this paper we carry out an algebraic investigation of the weak nilpotent minimum logic and its t-norm based axiomatic extensions. We consider the algebraic counterpart of WNM, the variety of WNM-algebras and prove that it is locally finite, so all its subvarieties are generated by finite chains. We give criteria to compare varieties generated by finite families of WNM-chains, in particular varieties generated by standard WNM-chains, or equivalently t-norm based axiomatic extensions of WNM, and we study their standard completeness (...)
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  7. Maximality in finite-valued Lukasiewicz logics defined by order filters.Marcelo E. Coniglio, Francesc Esteva, Joan Gispert & Lluis Godo - 2019 - Journal of Logic and Computation 29 (1):125-156.
  8.  19
    Bounded BCK‐algebras and their generated variety.Joan Gispert & Antoni Torrens - 2007 - Mathematical Logic Quarterly 53 (2):206-213.
    In this paper we prove that the equational class generated by bounded BCK-algebras is the variety generated by the class of finite simple bounded BCK-algebras. To obtain these results we prove that every simple algebra in the equational class generated by bounded BCK-algebras is also a relatively simple bounded BCK-algebra. Moreover, we show that every simple bounded BCK-algebra can be embedded into a simple integral commutative bounded residuated lattice. We extend our main results to some richer subreducts of the class (...)
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  9.  22
    Universal Classes of MV-Chains with Applications to Many-valued Logics.Joan Gispert - 2002 - Mathematical Logic Quarterly 48 (4):582-601.
    In this paper we characterize, classify and axiomatize all universal classes of MV-chains. Moreover, we accomplish analogous characterization, classification and axiomatization for congruence distributive quasivarieties of MV-algebras. Finally, we apply those results to study some finitary extensions of the Łukasiewicz infinite valued propositional calculus.
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  10.  10
    Finitary Extensions of the Nilpotent Minimum Logic and (Almost) Structural Completeness.Joan Gispert - 2018 - Studia Logica 106 (4):789-808.
    In this paper we study finitary extensions of the nilpotent minimum logic or equivalently quasivarieties of NM-algebras. We first study structural completeness of NML, we prove that NML is hereditarily almost structurally complete and moreover NM\, the axiomatic extension of NML given by the axiom \^{2}\leftrightarrow ^{2})^{2}\), is hereditarily structurally complete. We use those results to obtain the full description of the lattice of all quasivarieties of NM-algebras which allow us to characterize and axiomatize all finitary extensions of NML.
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  11.  30
    Axiomatic Extensions of IMT3 Logic.Joan Gispert & Antoni Torrens - 2005 - Studia Logica 81 (3):311-324.
    In this paper we characterize, classify and axiomatize all axiomatic extensions of the IMT3 logic. This logic is the axiomatic extension of the involutive monoidal t-norm logic given by ¬φ3 ∨ φ. For our purpose we study the lattice of all subvarieties of the class IMT3, which is the variety of IMTL-algebras given by the equation ¬(x 3) ∨ x ≈ ⊤, and it is the algebraic counterpart of IMT3 logic. Since every subvariety of IMT3 is generated by their totally (...)
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  12.  14
    Structural Completeness in Many-Valued Logics with Rational Constants.Joan Gispert, Zuzana Haniková, Tommaso Moraschini & Michał Stronkowski - 2022 - Notre Dame Journal of Formal Logic 63 (3):261-299.
    The logics RŁ, RP, and RG have been obtained by expanding Łukasiewicz logic Ł, product logic P, and Gödel–Dummett logic G with rational constants. We study the lattices of extensions and structural completeness of these three expansions, obtaining results that stand in contrast to the known situation in Ł, P, and G. Namely, RŁ is hereditarily structurally complete. RP is algebraized by the variety of rational product algebras that we show to be Q-universal. We provide a base of admissible rules (...)
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  13.  5
    Algebraic Expansions of Logics.Miguel Campercholi, Diego Nicolás Castaño, José Patricio Díaz Varela & Joan Gispert - 2023 - Journal of Symbolic Logic 88 (1):74-92.
    An algebraically expandable (AE) class is a class of algebraic structures axiomatizable by sentences of the form $\forall \exists! \mathop{\boldsymbol {\bigwedge }}\limits p = q$. For a logic L algebraized by a quasivariety $\mathcal {Q}$ we show that the AE-subclasses of $\mathcal {Q}$ correspond to certain natural expansions of L, which we call algebraic expansions. These turn out to be a special case of the expansions by implicit connectives studied by X. Caicedo. We proceed to characterize all the AE-subclasses of (...)
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  14.  10
    Degree-Preserving Gödel Logics with an Involution: Intermediate Logics and Paraconsistency.Marcelo E. Coniglio, Francesc Esteva, Joan Gispert & Lluis Godo - 2021 - In Ofer Arieli & Anna Zamansky (eds.), Arnon Avron on Semantics and Proof Theory of Non-Classical Logics. Springer Verlag. pp. 107-139.
    In this paper we study intermediate logics between the logic G≤∼, the degree preserving companion of Gödel fuzzy logic with involution G∼ and classical propositional logic CPL, as well as the intermediate logics of their finite-valued counterparts G≤n∼. Although G≤∼ and G≤ are explosive w.r.t. Gödel negation ¬, they are paraconsistent w.r.t. the involutive negation ∼. We introduce the notion of saturated paraconsistency, a weaker notion than ideal paraconsistency, and we fully characterize the ideal and the saturated paraconsistent logics between (...)
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  15.  12
    Lattice BCK logics with Modus Ponens as unique rule.Joan Gispert & Antoni Torrens - 2014 - Mathematical Logic Quarterly 60 (3):230-238.
    Lattice BCK logic is the expansion of the well known Meredith implicational logic BCK expanded with lattice conjunction and disjunction. Although its natural axiomatization has three rules named modus ponens, ∨‐rule and ∧‐rule, we show that we can give an equivalent presentation with just modus ponens and ∧‐rule, however it is impossible to obtain an equivalent presentation with modus ponens as unique rule. In this paper we study and characterize all axiomatic extensions of lattice BCK logic with modus ponens as (...)
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  16.  24
    Quasivarieties generated by simple MV-algebras.Joan Gispert & Antoni Torrens - 1998 - Studia Logica 61 (1):79-99.
    In this paper we show that the quasivariety generated by an infinite simple MV-algebra only depends on the rationals which it contains. We extend this property to arbitrary families of simple MV-algebras.
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