Quasi-subtractive varieties

Journal of Symbolic Logic 76 (4):1261-1286 (2011)


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 A of the τ-assertional logic of . Moreover, if has a constant 1 in its type and is 1-subtractive, the deductive filters on A ∈ of the 1-assertional logic of coincide with the -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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Francesco Paoli
Universita di Cagliari

References found in this work

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Equivalential logics.Janusz Czelakowski - 1981 - Studia Logica 40 (3):227-236.
Topology and Duality in Modal Logic.Giovanni Sambin & Virginia Vaccaro - 1988 - Annals of Pure and Applied Logic 37 (3):249-296.

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