Quasi-subtractive varieties
Tomasz Kowalski, Francesco Paoli, Matthew Spinks · Journal of Symbolic Logic · 2011
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 memberAof aτ-regular variety the lattice of congruences ofAis isomorphic to the lattice of deductive filters onAof theτ-assertional logic of . Moreover, if has a constant 1 in its type and is 1-subtractive, the deductive filters onA∈ of the 1-assertional logic of coincide with the -ideals ofAin 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 andτ-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.