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