On subtractive varieties. I (Q1319053)

A variety \({\mathcal V}\) (or an algebra \({\mathbf A})\) is said to be subtractive if it satisfies the identities \(s(x,x) \approx 0\) and \(s(x,0) \approx x\) for some binary term \(s\) and some nullary term 0. Such an \(s\) is called a subtraction term. Let \({\mathcal K}\) be a class of similar algebras, then a term \(p({\mathbf x}, {\mathbf y}) = p(x_ 1, \dots, x_ m, y_ 1, \dots, y_ n)\) is a \({\mathcal K}\)-ideal term if the identity \(p({\mathbf x}, \mathbf{0}) \approx 0\) holds in \({\mathcal K}\) (we then write \(p({\mathbf x}, {\mathbf y})\in \text{IT}_{\mathcal K} ({\mathbf y}))\). An nonempty subset \(I\) of \({\mathbf A} \in {\mathcal K}\) is a \({\mathcal K}\)-ideal of \({\mathbf A}\) if for any \({\mathbf a} \in {\mathbf A}\) and \({\mathbf b} \in I\), \(p({\mathbf a},{\mathbf b}) \in I\). We write \(I({\mathbf A})\) instead of \(\text{I}_{\{{\mathbf A}\}} ({\mathbf A)}\) and let \(N({\mathbf A})\) denote \(\{ [0]_ \theta \mid \theta \in \text{Con} ({\mathbf A})\}\). Although subtractive varieties are not in general congruence modular, their congruences are 0-permutable. Moreover, if \({\mathbf A} \in {\mathcal V}\), a subtractive variety, then \(\text{I}_{\mathcal V} ({\mathbf A}) = \text{I} ({\mathbf A}) = \text{N} ({\mathbf A})\). Also, if \({\mathbf A}\) is subtractive, \(\text{I} ({\mathbf A})\) is modular. These facts enable a meaningful commutator to be defined for subtractive varieties, using ideals rather than congruences. This paper contains many interesting results about such commutators.