M-Sets and the Representation Problem

Studia Logica 103 (1):21-51 (2015)
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Abstract

The “representation problem” in abstract algebraic logic is that of finding necessary and sufficient conditions for a structure, on a well defined abstract framework, to have the following property: that for every structural closure operator on it, every structural embedding of the expanded lattice of its closed sets into that of the closed sets of another structural closure operator on another similar structure is induced by a structural transformer between the base structures. This question arose from Blok and Jónsson abstract analysis of one of Blok and Pigozzis’s characterizations of algebraizable logics. The problem, which was later on reformulated independently by Gil-Férez and by Galatos and Tsinakis, was solved by Galatos and Tsinakis in the more abstract framework of the category of modules over a complete residuated lattice, and by Galatos and Gil-Férez in the even more abstract setting of modules over a quantaloid. We solve the representation problem in Blok and Jónsson’s original context of M-sets, where M is a monoid, and characterise the corresponding M-sets both in categorical terms and in terms of their inner structure, using the notions of a graded M-set and a generalized variable introduced by Gil-Férez

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Logics of variable inclusion and the lattice of consequence relations.Michele Pra Baldi - 2020 - Journal of Applied Non-Classical Logics 30 (4):367-381.
The semantic isomorphism theorem in abstract algebraic logic.Tommaso Moraschini - 2016 - Annals of Pure and Applied Logic 167 (12):1298-1331.

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References found in this work

Equivalence of Consequence Operations.W. J. Blok & Bjarni Jónsson - 2006 - Studia Logica 83 (1-3):91-110.
Correspondences between Gentzen and Hilbert Systems.J. G. Raftery - 2006 - Journal of Symbolic Logic 71 (3):903 - 957.
Representations of structural closure operators.José Gil-Férez - 2011 - Archive for Mathematical Logic 50 (1-2):45-73.

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