On Subtractive Varieties, I

A variety V is subtractive if it obeys the laws s(x, x) = 0, s(x, 0) = x for some binary term s and constant 0. This means that V has 0-permutable congruences (namely [0]R circle S = [0]S circle R for any congruences R, S of any algebra in V). We present the basic features of such varieties, mainly from the viewpoint of ideal theory. Subtractivity does not imply congruence modularity, yet the commutator theory for ideals works fine. We characterize i-Abelian algebras, (i.e. those in which the commutator is identically 0). In the appendix we consider the case of a ''classical'' ideal theory (comprising: groups, loops, rings, Heyting and Boolean algebras, even with multioperators and virtually all algebras coming from logic) and we characterize the corresponding class of subtractive varieties.