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Products of classes of residuated structures

Bjarni Jónsson & Constantine Tsinakis

Studia Logica 77 (2):267 - 292 (2004)

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Algebraic Semantics for a Mixed Type Fragment of IPC.Eryk Lipka & Katarzyna Słomczyńska - forthcoming - Studia Logica:1-25.details We investigate algebraically the fragment of the intuitionistic propositional calculus consisting of equivalence together with conjunction on the intuitionistic regularizations. We find that this fragment is strongly algebraizable with the equivalent algebraic semantics being the variety of equivalential algebras with an additional binary operation that can be interpreted as the meet on regular elements. We give a finite equational base for this variety, and investigate its properties, in particular the commutator. As applications, we prove that the fragment is hereditarily structurally (...) Download Export citation Bookmark
On Birkhoff’s Common Abstraction Problem.F. Paoli & C. Tsinakis - 2012 - Studia Logica 100 (6):1079-1105.details In his milestone textbook Lattice Theory, Garrett Birkhoff challenged his readers to develop a "common abstraction" that includes Boolean algebras and lattice-ordered groups as special cases. In this paper, after reviewing the past attempts to solve the problem, we provide our own answer by selecting as common generalization of �� and �� their join ��∨�� in the lattice of subvarieties of ��ℒ (the variety of FL-algebras); we argue that such a solution is optimal under several respects and we give an (...) Download Export citation Bookmark
Non-commutative logical algebras and algebraic quantales.Wolfgang Rump & Yi Chuan Yang - 2014 - Annals of Pure and Applied Logic 165 (2):759-785.details Quantum B-algebras, the partially ordered implicational algebras arising as subreducts of quantales, are introduced axiomatically. It is shown that they provide a unified semantic for non-commutative algebraic logic. Specifically, they cover the vast majority of implicational algebras like BCK-algebras, residuated lattices, partially ordered groups, BL- and MV-algebras, effect algebras, and their non-commutative extensions. The opposite of the category of quantum B-algebras is shown to be equivalent to the category of logical quantales, in the way that every quantum B-algebra admits a (...) Download Export citation Bookmark 5 citations
Quasi-subtractive varieties.Tomasz Kowalski, Francesco Paoli & Matthew Spinks - 2011 - Journal of Symbolic Logic 76 (4):1261-1286.details Varieties like groups, rings, or Boolean algebras have the property that, in any of their members, the lattice of congruences is isomorphic to a lattice of more manageable objects, for example normal subgroups of groups, two-sided ideals of rings, filters (or ideals) of Boolean algebras.algebraic logic can explain these phenomena at a rather satisfactory level of generality: in every member A of a τ-regular variety �� the lattice of congruences of A is isomorphic to the lattice of deductive filters on (...) Download Export citation Bookmark 2 citations
Minimal Varieties of Involutive Residuated Lattices.Constantine Tsinakis & Annika M. Wille - 2006 - Studia Logica 83 (1-3):407-423.details We establish the existence uncountably many atoms in the subvariety lattice of the variety of involutive residuated lattices. The proof utilizes a construction used in the proof of the corresponding result for residuated lattices and is based on the fact that every residuated lattice with greatest element can be associated in a canonical way with an involutive residuated lattice. Download Export citation Bookmark 12 citations

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