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Quasi-subtractive varieties

Published online by Cambridge University Press:  12 March 2014

Tomasz Kowalski
Affiliation:
Department of Mathematics and Statistics, University of Melbourne, Melbourne, Australia, E-mail: tomaszk@unimelb.edu.au
Francesco Paoli
Affiliation:
Department of Education, University of Cagliari, Cagliari, Italy, E-mail: paoli@unica.it
Matthew Spinks
Affiliation:
Department of Education, University of Cagliari, Cagliari, Italy, E-mail: mspinksau@yahoo.com.au

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 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.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2011

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References

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