12 found
  1.  40
    Residuated Lattices: An Algebraic Glimpse at Substructural Logics.Nikolaos Galatos, Peter Jipsen, Tomasz Kowalski & Hiroakira Ono - 2007 - Elsevier.
    This is also where we begin investigating lattices of logics and varieties, rather than particular examples.
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  2.  63
    Algebraic Proof Theory for Substructural Logics: Cut-Elimination and Completions.Agata Ciabattoni, Nikolaos Galatos & Kazushige Terui - 2012 - Annals of Pure and Applied Logic 163 (3):266-290.
  3.  67
    Algebraization, Parametrized Local Deduction Theorem and Interpolation for Substructural Logics Over FL.Nikolaos Galatos & Hiroakira Ono - 2006 - Studia Logica 83 (1-3):279-308.
    Substructural logics have received a lot of attention in recent years from the communities of both logic and algebra. We discuss the algebraization of substructural logics over the full Lambek calculus and their connections to residuated lattices, and establish a weak form of the deduction theorem that is known as parametrized local deduction theorem. Finally, we study certain interpolation properties and explain how they imply the amalgamation property for certain varieties of residuated lattices.
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  4.  53
    Equivalence of Consequence Relations: An Order-Theoretic and Categorical Perspective.Nikolaos Galatos & Constantine Tsinakis - 2009 - Journal of Symbolic Logic 74 (3):780-810.
    Equivalences and translations between consequence relations abound in logic. The notion of equivalence can be defined syntactically, in terms of translations of formulas, and order-theoretically, in terms of the associated lattices of theories. W. Blok and D. Pigozzi proved in [4] that the two definitions coincide in the case of an algebraizable sentential deductive system. A refined treatment of this equivalence was provided by W. Blok and B. Jónsson in [3]. Other authors have extended this result to the cases of (...)
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  5.  30
    Adding Involution to Residuated Structures.Nikolaos Galatos & James G. Raftery - 2004 - Studia Logica 77 (2):181 - 207.
    Two constructions for adding an involution operator to residuated ordered monoids are investigated. One preserves integrality and the mingle axiom x 2x but fails to preserve the contraction property xx 2. The other has the opposite preservation properties. Both constructions preserve commutativity as well as existent nonempty meets and joins and self-dual order properties. Used in conjunction with either construction, a result of R.T. Brady can be seen to show that the equational theory of commutative distributive residuated lattices (without involution) (...)
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  6.  19
    Algebraic Proof Theory: Hypersequents and Hypercompletions.Agata Ciabattoni, Nikolaos Galatos & Kazushige Terui - 2017 - Annals of Pure and Applied Logic 168 (3):693-737.
  7.  23
    Cut Elimination and Strong Separation for Substructural Logics: An Algebraic Approach.Nikolaos Galatos & Hiroakira Ono - 2010 - Annals of Pure and Applied Logic 161 (9):1097-1133.
    We develop a general algebraic and proof-theoretic study of substructural logics that may lack associativity, along with other structural rules. Our study extends existing work on substructural logics over the full Lambek Calculus [34], Galatos and Ono [18], Galatos et al. [17]). We present a Gentzen-style sequent system that lacks the structural rules of contraction, weakening, exchange and associativity, and can be considered a non-associative formulation of . Moreover, we introduce an equivalent Hilbert-style system and show that the logic associated (...)
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  8.  36
    Glivenko Theorems for Substructural Logics Over FL.Nikolaos Galatos & Hiroakira Ono - 2006 - Journal of Symbolic Logic 71 (4):1353 - 1384.
    It is well known that classical propositional logic can be interpreted in intuitionistic propositional logic. In particular Glivenko's theorem states that a formula is provable in the former iff its double negation is provable in the latter. We extend Glivenko's theorem and show that for every involutive substructural logic there exists a minimum substructural logic that contains the first via a double negation interpretation. Our presentation is algebraic and is formulated in the context of residuated lattices. In the last part (...)
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  9.  39
    Equational Bases for Joins of Residuated-Lattice Varieties.Nikolaos Galatos - 2004 - Studia Logica 76 (2):227 - 240.
    Given a positive universal formula in the language of residuated lattices, we construct a recursive basis of equations for a variety, such that a subdirectly irreducible residuated lattice is in the variety exactly when it satisfies the positive universal formula. We use this correspondence to prove, among other things, that the join of two finitely based varieties of commutative residuated lattices is also finitely based. This implies that the intersection of two finitely axiomatized substructural logics over FL + is also (...)
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  10.  19
    Preface.Nikolaos Galatos, Peter Jipsen & Hiroakira Ono - 2012 - Studia Logica 100 (6):1059-1062.
  11.  7
    Proof Theory for Lattice-Ordered Groups.Nikolaos Galatos & George Metcalfe - 2016 - Annals of Pure and Applied Logic 167 (8):707-724.
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  12. Twist Structures and Nelson Conuclei.Manuela Busaniche, Nikolaos Galatos & Miguel Andrés Marcos - forthcoming - Studia Logica:1-39.
    Motivated by Kalman residuated lattices, Nelson residuated lattices and Nelson paraconsistent residuated lattices, we provide a natural common generalization of them. Nelson conucleus algebras unify these examples and further extend them to the non-commutative setting. We study their structure, establish a representation theorem for them in terms of twist structures and conuclei that results in a categorical adjunction, and explore situations where the representation is actually an isomorphism. In the latter case, the adjunction is elevated to a categorical equivalence. By (...)
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