24 found
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  1.  21
    Free Algebras in Varieties of Glivenko MTL-Algebras Satisfying the Equation 2(X²) = (2x)².Roberto Cignoli & Antoni Torrens Torrell - 2006 - Studia Logica 83 (1-3):157 - 181.
    The aim of this paper is to give a description of the free algebras in some varieties of Glivenko MTL-algebras having the Boolean retraction property. This description is given in terms of weak Boolean products over Cantor spaces. We prove that in some cases the stalks can be obtained in a constructive way from free kernel DL-algebras, which are the maximal radical of directly indecomposable Glivenko MTL-algebras satisfying the equation in the title. We include examples to show how we can (...)
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  2.  79
    Proper N-Valued Łukasiewicz Algebras as s-Algebras of Łukasiewicz N-Valued Prepositional Calculi.Roberto Cignoli - 1982 - Studia Logica 41 (1):3 - 16.
    Proper n-valued ukasiewicz algebras are obtained by adding some binary operators, fulfilling some simple equations, to the fundamental operations of n-valued ukasiewicz algebras. They are the s-algebras corresponding to an axiomatization of ukasiewicz n-valued propositional calculus that is an extention of the intuitionistic calculus.
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  3.  58
    An Algebraic Approach to Intuitionistic Connectives.Xavier Caicedo & Roberto Cignoli - 2001 - Journal of Symbolic Logic 66 (4):1620-1636.
    It is shown that axiomatic extensions of intuitionistic propositional calculus defining univocally new connectives, including those proposed by Gabbay, are strongly complete with respect to valuations in Heyting algebras with additional operations. In all cases, the double negation of such a connective is equivalent to a formula of intuitionistic calculus. Thus, under the excluded third law it collapses to a classical formula, showing that this condition in Gabbay's definition is redundant. Moreover, such connectives can not be interpreted in all Heyting (...)
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  4.  13
    Glivenko Like Theorems in Natural Expansions of BCK‐Logic.Roberto Cignoli & Antoni Torrens Torrell - 2004 - Mathematical Logic Quarterly 50 (2):111-125.
    The classical Glivenko theorem asserts that a propositional formula admits a classical proof if and only if its double negation admits an intuitionistic proof. By a natural expansion of the BCK-logic with negation we understand an algebraizable logic whose language is an expansion of the language of BCK-logic with negation by a family of connectives implicitly defined by equations and compatible with BCK-congruences. Many of the logics in the current literature are natural expansions of BCK-logic with negation. The validity of (...)
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  5.  15
    Hájek Basic Fuzzy Logic and Łukasiewicz Infinite-Valued Logic.Roberto Cignoli & Antoni Torrens - 2003 - Archive for Mathematical Logic 42 (4):361-370.
    . Using the theory of BL-algebras, it is shown that a propositional formula ϕ is derivable in Łukasiewicz infinite valued Logic if and only if its double negation ˜˜ϕ is derivable in Hájek Basic Fuzzy logic. If SBL is the extension of Basic Logic by the axiom ) → ψ, then ϕ is derivable in in classical logic if and only if ˜˜ ϕ is derivable in SBL. Axiomatic extensions of Basic Logic are in correspondence with subvarieties of the variety (...)
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  6.  10
    Erratum To: Free Algebras in Varieties of Glivenko MTL-Algebras Satisfying the Equation $${2(X^2) = (2x)^2}$$ 2 ( X 2 ) = ( 2 X ) 2.Antoni Torrens & Roberto Cignoli - 2017 - Studia Logica 105 (1):227-228.
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  7.  36
    An Elementary Proof of Chang's Completeness Theorem for the Infinite-Valued Calculus of Lukasiewicz.Roberto Cignoli & Daniele Mundici - 1997 - Studia Logica 58 (1):79-97.
    The interpretation of propositions in Lukasiewicz's infinite-valued calculus as answers in Ulam's game with lies--the Boolean case corresponding to the traditional Twenty Questions game--gives added interest to the completeness theorem. The literature contains several different proofs, but they invariably require technical prerequisites from such areas as model-theory, algebraic geometry, or the theory of ordered groups. The aim of this paper is to provide a self-contained proof, only requiring the rudiments of algebra and convexity in finite-dimensional vector spaces.
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  8.  40
    Varieties of Commutative Integral Bounded Residuated Lattices Admitting a Boolean Retraction Term.Roberto Cignoli & Antoni Torrens - 2012 - Studia Logica 100 (6):1107-1136.
    Let ${\mathbb{BRL}}$ denote the variety of commutative integral bounded residuated lattices (bounded residuated lattices for short). A Boolean retraction term for a subvariety ${\mathbb{V}}$ of ${\mathbb{BRL}}$ is a unary term t in the language of bounded residuated lattices such that for every ${{\bf A} \in \mathbb{V}, t^{A}}$ , the interpretation of the term on A, defines a retraction from A onto its Boolean skeleton B(A). It is shown that Boolean retraction terms are equationally definable, in the sense that there is (...)
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  9.  31
    An Elementary Presentation of the Equivalence Between MV-Algebras and L-Groups with Strong Unit.Roberto Cignoli & Daniele Mundici - 1998 - Studia Logica 61 (1):49-64.
    Aim of this paper is to provide a self-contained presentation of the natural equivalence between MV-algebras and lattice-ordered abelian groups with strong unit.
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  10.  9
    Commutative Integral Bounded Residuated Lattices with an Added Involution.Roberto Cignoli & Francesc Esteva - 2010 - Annals of Pure and Applied Logic 161 (2):150-160.
    A symmetric residuated lattice is an algebra such that is a commutative integral bounded residuated lattice and the equations x=x and =xy are satisfied. The aim of the paper is to investigate the properties of the unary operation ε defined by the prescription εx=x→0. We give necessary and sufficient conditions for ε being an interior operator. Since these conditions are rather restrictive →0)=1 is satisfied) we consider when an iteration of ε is an interior operator. In particular we consider the (...)
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  11.  14
    Glivenko Like Theorems in Natural Expansions of BCK-Logic.Roberto Cignoli & Antoni Torrell - 2004 - Mathematical Logic Quarterly 50 (2):111-125.
    The classical Glivenko theorem asserts that a propositional formula admits a classical proof if and only if its double negation admits an intuitionistic proof. By a natural expansion of the BCK-logic with negation we understand an algebraizable logic whose language is an expansion of the language of BCK-logic with negation by a family of connectives implicitly defined by equations and compatible with BCK-congruences. Many of the logics in the current literature are natural expansions of BCK-logic with negation. The validity of (...)
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  12.  38
    Free Q-Distributive Lattices.Roberto Cignoli - 1996 - Studia Logica 56 (1-2):23 - 29.
    The dual spaces of the free distributive lattices with a quantifier are constructed, generalizing Halmos' construction of the dual spaces of free monadic Boolean algebras.
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  13.  9
    An Algebraic Approach to Intuitionistic Connectives.Xavier Caicedo & Roberto Cignoli - 2001 - Journal of Symbolic Logic 66 (4):1620-1636.
    It is shown that axiomatic extensions of intuitionistic propositional calculus defining univocally new connectives, including those proposed by Gabbay, are strongly complete with respect to valuations in Heyting algebras with additional operations. In all cases, the double negation of such a connective is equivalent to a formula of intuitionistic calculus. Thus, under the excluded third law it collapses to a classical formula, showing that this condition in Gabbay's definition is redundant. Moreover, such connectives can not be interpreted in all Heyting (...)
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  14.  22
    Coproducts in the Categories of Kleene and Three-Valued Łukasiewicz Algebras.Roberto Cignoli - 1979 - Studia Logica 38 (3):237 - 245.
    It is given an explicit description of coproducts in the category of Kleene algebras in terms of the dual topological spaces. As an application, a description of dual spaces of free Kleene algebras is given. It is also shown that the coproduct of a family of three-valued ukasiewicz algebras in the category of Kleene algebras is the same as the coproduct in the subcategory of three-valued ukasiewicz algebras.
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  15.  22
    A Categorical Equivalence for Stonean Residuated Lattices.Manuela Busaniche, Roberto Cignoli & Miguel Andrés Marcos - 2019 - Studia Logica 107 (2):399-421.
    We follow the ideas given by Chen and Grätzer to represent Stone algebras and adapt them for the case of Stonean residuated lattices. Given a Stonean residuated lattice, we consider the triple formed by its Boolean skeleton, its algebra of dense elements and a connecting map. We define a category whose objects are these triples and suitably defined morphisms, and prove that we have a categorical equivalence between this category and that of Stonean residuated lattices. We compare our results with (...)
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  16.  18
    Remarks on an Algebraic Semantics for Paraconsistent Nelson's Logic.Manuela Busaniche & Roberto Cignoli - 2011 - Manuscrito 34 (1):99-114.
    In the paper Busaniche and Cignoli we presented a quasivariety of commutative residuated lattices, called NPc-lattices, that serves as an algebraic semantics for paraconsistent Nelson’s logic. In the present paper we show that NPc-lattices form a subvariety of the variety of commutative residuated lattices, we study congruences of NPc-lattices and some subvarieties of NPc-lattices.
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  17.  8
    An Algebraic Approach to Elementary Theories Based on N‐Valued Lukasiewicz Logics.Roberto Cignoli - 1984 - Mathematical Logic Quarterly 30 (1‐6):87-96.
  18.  27
    An Algebraic Approach to Elementary Theories Based Onn-Valued Lukasiewicz Logics.Roberto Cignoli - 1984 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 30 (1-6):87-96.
  19. Algebraic Foundations of Many-Valued Reasoning.Roberto Cignoli - 1999 - Kluwer Academic Publishers.
    This unique textbook states and proves all the major theorems of many-valued propositional logic and provides the reader with the most recent developments and trends, including applications to adaptive error-correcting binary search. The book is suitable for self-study, making the basic tools of many-valued logic accessible to students and scientists with a basic mathematical knowledge who are interested in the mathematical treatment of uncertain information. Stressing the interplay between algebra and logic, the book contains material never before published, such as (...)
     
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  20.  38
    Ayda Ignez Arruda (1936–1983).Roberto Cignoli - 1984 - Studia Logica 43 (1-2):1 - 2.
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  21.  10
    Boolean Skeletons of MV-Algebras and ℓ-Groups.Roberto Cignoli - 2011 - Studia Logica 98 (1-2):141-147.
    Let Γ be Mundici’s functor from the category $${\mathcal{LG}}$$ whose objects are the lattice-ordered abelian groups ( ℓ -groups for short) with a distinguished strong order unit and the morphisms are the unital homomorphisms, onto the category $${\mathcal{MV}}$$ of MV-algebras and homomorphisms. It is shown that for each strong order unit u of an ℓ -group G , the Boolean skeleton of the MV-algebra Γ ( G , u ) is isomorphic to the Boolean algebra of factor congruences of G.
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  22.  33
    Complete and Atomic Algebras of the Infinite Valued Łukasiewicz Logic.Roberto Cignoli - 1991 - Studia Logica 50 (3-4):375 - 384.
    The infinite-valued logic of ukasiewicz was originally defined by means of an infinite-valued matrix. ukasiewicz took special forms of negation and implication as basic connectives and proposed an axiom system that he conjectured would be sufficient to derive the valid formulas of the logic; this was eventually verified by M. Wajsberg. The algebraic counterparts of this logic have become know as Wajsberg algebras. In this paper we show that a Wajsberg algebra is complete and atomic (as a lattice) if and (...)
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  23.  36
    Maximal Subalgebras of MVn-Algebras. A Proof of a Conjecture of A. Monteiro.Roberto Cignoli & Luiz Monteiro - 2006 - Studia Logica 84 (3):393 - 405.
    For each integer n ≥ 2, MVn denotes the variety of MV-algebras generated by the MV-chain with n elements. Algebras in MVn are represented as continuous functions from a Boolean space into a n-element chain equipped with the discrete topology. Using these representations, maximal subalgebras of algebras in MVn are characterized, and it is shown that proper subalgebras are intersection of maximal subalgebras. When A ∈ MV3, the mentioned characterization of maximal subalgebras of A can be given in terms of (...)
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  24.  13
    Maximal Subalgebras of MVn-Algebras. A Proof of a Conjecture of A. Monteiro.Roberto Cignoli & Luiz Monteiro - 2006 - Studia Logica 84 (3):393-405.
    For each integer n ≥ 2, MVn denotes the variety of MV-algebras generated by the MV-chain with n elements. Algebras in MVn are represented as continuous functions from a Boolean space into a n-element chain equipped with the discrete topology. Using these representations, maximal subalgebras of algebras in MVn are characterized, and it is shown that proper subalgebras are intersection of maximal subalgebras. When A ∈ MV3, the mentioned characterization of maximal subalgebras of A can be given in terms of (...)
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