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Equivalence of consequence relations: an order-theoretic and categorical perspective

Published online by Cambridge University Press:  12 March 2014

Nikolaos Galatos  and
Constantine Tsinakis
Nikolaos Galatos
Affiliation:
Department of Mathematics, University of Denver, 2360 S. Gaylord St. Denver, Co 80208, USA, E-mail: ngalatos@du.edu, URL: http://www.math.du.edu/~ngalatos
Constantine Tsinakis
Affiliation:
Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, Tn 37240, USA, E-mail: constantine.tsinakis@vanderbilt.edu, URL: http://www.math.vanderbilt.edu/people/tsinakis
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Abstract

Equivalences and translations between consequence relations abound in logic. The notion of equivalence can be denned 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 κ-deductive systems and of consequence relations on associative, commutative, multiple conclusion sequents. Our main result subsumes all existing results in the literature and reveals their common character. The proofs are of order-theoretic and categorical nature.

Keywords

Consequence relationclosure operatorequivalencealgebraizableresiduated latticemoduleprojective
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 3 , September 2009 , pp. 780 - 810
Copyright
Copyright © Association for Symbolic Logic 2009

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