17 found
Order:
  1.  76
    On the Infinite-Valued Łukasiewicz Logic That Preserves Degrees of Truth.Josep Maria Font, Àngel J. Gil, Antoni Torrens & Ventura Verdú - 2006 - Archive for Mathematical Logic 45 (7):839-868.
    Łukasiewicz’s infinite-valued logic is commonly defined as the set of formulas that take the value 1 under all evaluations in the Łukasiewicz algebra on the unit real interval. In the literature a deductive system axiomatized in a Hilbert style was associated to it, and was later shown to be semantically defined from Łukasiewicz algebra by using a “truth-preserving” scheme. This deductive system is algebraizable, non-selfextensional and does not satisfy the deduction theorem. In addition, there exists no Gentzen calculus fully adequate (...)
    Direct download (6 more)  
     
    Export citation  
     
    Bookmark   12 citations  
  2.  16
    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 (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   8 citations  
  3.  27
    W-Algebras Which Are Boolean Products of Members of SR[1] and CW-Algebras.Antoni Torrens - 1987 - Studia Logica 46 (3):265 - 274.
    We show that the class of all isomorphic images of Boolean Products of members of SR [1] is the class of all archimedean W-algebras. We obtain this result from the characterization of W-algebras which are isomorphic images of Boolean Products of CW-algebras.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   7 citations  
  4.  16
    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 (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   5 citations  
  5.  11
    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.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  6.  44
    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 (...)
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  7.  18
    Lukasiewicz Logic and Wajsberg Algebras.Antonio J. Rodriguez, Antoni Torrens & Ventura Verdú - 1990 - Bulletin of the Section of Logic 19 (2):51-55.
  8.  62
    Wajsberg Algebras and Post Algebras.Antonio Jesús Rodríguez & Antoni Torrens - 1994 - Studia Logica 53 (1):1 - 19.
    We give a presentation of Post algebras of ordern+1 (n1) asn+1 bounded Wajsberg algebras with an additional constant, and we show that a Wajsberg algebra admits a P-algebra reduct if and only if it isn+1 bounded.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  9.  15
    On The Role of The Polynomial (X → Y) → Y in Some Implicative Algebras.Antoni Torrens - 1988 - Mathematical Logic Quarterly 34 (2):117-122.
  10.  24
    An Approach to Glivenko’s Theorem in Algebraizable Logics.Antoni Torrens - 2008 - Studia Logica 88 (3):349-383.
    In a classical paper [15] V. Glivenko showed that a proposition is classically demonstrable if and only if its double negation is intuitionistically demonstrable. This result has an algebraic formulation: the double negation is a homomorphism from each Heyting algebra onto the Boolean algebra of its regular elements. Versions of both the logical and algebraic formulations of Glivenko’s theorem, adapted to other systems of logics and to algebras not necessarily related to logic can be found in the literature (see [2, (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  11.  27
    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 (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  12.  21
    On The Role of The Polynomial →Y in Some Implicative Algebras.Antoni Torrens - 1988 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 34 (2):117-122.
  13.  8
    Lattice BCK Logics with Modus Ponens as Unique Rule.Joan Gispert & Antoni Torrens - 2014 - Mathematical Logic Quarterly 60 (3):230-238.
  14.  20
    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.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  15.  18
    Cyclic Elements in MV‐Algebras and Post Algebras.Antoni Torrens - 1994 - Mathematical Logic Quarterly 40 (4):431-444.
    In this paper we characterize the MV-algebras containing as subalgebras Post algebras of finitely many orders. For this we study cyclic elements in MV-algebras which are the generators of the fundamental chain of the Post algebras.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  16.  21
    On the Definability of Join by Means of Polynomials in Implicative Algebras.Antoni Torrens - 1985 - Bulletin of the Section of Logic 14 (4):158-162.
    In this paper we see that the answer of this question is affirmative. We prove this for Dco-algebras and as special case we obtain the result for Positive Implication algebras. First we give, without proof, the properties of Dco-algebras and S-algebras and their connection with Positive Implication algebras and Implication algebras. These results can be found in [T] and [IT].
    Direct download  
     
    Export citation  
     
    Bookmark  
  17.  13
    Semisimples in Varieties of Commutative Integral Bounded Residuated Lattices.Antoni Torrens - 2016 - Studia Logica 104 (5):849-867.
    In any variety of bounded integral residuated lattice-ordered commutative monoids the class of its semisimple members is closed under isomorphic images, subalgebras and products, but it is not closed under homomorphic images, and so it is not a variety. In this paper we study varieties of bounded residuated lattices whose semisimple members form a variety, and we give an equational presentation for them. We also study locally representable varieties whose semisimple members form a variety. Finally, we analyze the relationship with (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark