Journal of Symbolic Logic

Quasi-subtractive varieties

Tomasz Kowalski, Francesco Paoli, and Matthew Spinks

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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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J. Symbolic Logic, Volume 76, Issue 4 (2011), 1261-1286.

First available in Project Euclid: 11 October 2011

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Kowalski, Tomasz; Paoli, Francesco; Spinks, Matthew. Quasi-subtractive varieties. J. Symbolic Logic 76 (2011), no. 4, 1261--1286. doi:10.2178/jsl/1318338848.

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