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December 2011 Quasi-subtractive varieties
Tomasz Kowalski, Francesco Paoli, Matthew Spinks
J. Symbolic Logic 76(4): 1261-1286 (December 2011). DOI: 10.2178/jsl/1318338848

Abstract

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. Abstract algebraic logic can explain these phenomena at a rather satisfactory level of generality: in every member A of a τ-regular variety 𝕍 the lattice of congruences of A is isomorphic to the lattice of deductive filters on A of the τ-assertional logic of 𝕍. Moreover, if 𝕍 has a constant 1 in its type and is 1-subtractive, the deductive filters on A∈ 𝕍 of the 1-assertional logic of 𝕍 coincide with the 𝕍-ideals of A in the sense of Gumm and Ursini, for which we have a manageable concept of ideal generation. However, there are isomorphism theorems, for example, in the theories of residuated lattices, pseudointerior algebras and quasi-MV algebras that cannot be subsumed by these general results. The aim of the present paper is to appropriately generalise the concepts of subtractivity and τ-regularity in such a way as to shed some light on the deep reason behind such theorems. The tools and concepts we develop hereby provide a common umbrella for the algebraic investigation of several families of logics, including substructural logics, modal logics, quantum logics, and logics of constructive mathematics.

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Tomasz Kowalski. Francesco Paoli. Matthew Spinks. "Quasi-subtractive varieties." J. Symbolic Logic 76 (4) 1261 - 1286, December 2011. https://doi.org/10.2178/jsl/1318338848

Information

Published: December 2011
First available in Project Euclid: 11 October 2011

zbMATH: 1254.03119
MathSciNet: MR2895395
Digital Object Identifier: 10.2178/jsl/1318338848

Rights: Copyright © 2011 Association for Symbolic Logic

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Vol.76 • No. 4 • December 2011
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