Quasi-subtractive varieties

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: e.g. normal subgroups of groups, two-sided ideals of rings, …lters (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 -regular variety V, in fact, the lattice of congruences of A is isomorphic to the lattice of deductive …lters on A of the -assertional logic of V. Moreover, if V has a constant 1 in its type and is 1-subtractive, the deductive …lters on A 2 V of the 1assertional logic of V coincide with the V-ideals ofA in the sense of Gumm and Ursini, for which we have a manageable concept of ideal generation. However, there are isomorphism theorems e.g. 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 -regularity in such a way as to shed some light on the deep reason behind such theorems, as well as (possibly) many more. 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, logics of constructive mathematics.