Categorical (binary) difference terms and protomodularity

Abstract

In this paper we obtain an intrinsic syntactical characterization of protomodularity, via so-called categorical difference terms, similar to the one known in the case of varieties involving binary terms d satisfying d(x, x) = d(y, y). We also show that purely categorical modifications of the condition in the characterization give characterizations of Mal'tsev and additive categories, thus revealing a new conceptual link between these three classes of categories, and hence, also between the corresponding classes of varieties.

Normalizers and Split Extensions

“…Recall a paragroup [11] is a set X equipped with a binary operation satisfying the cancellation rule (y/x)/(z/x) = (y/z) as well as x/(x/x) = x and x/x = y/y. A non empty paragroup X determines a group structure defined by x/x = 1 and x.y = x/(1/y).…”

Approximate Hagemann–Mitschke Co-operations

“…We believe this aspect of our work gives a good illustration of the strength and generality of D. Bourn and Z. Janelidze's technique. See also [4,3,10,5] where the authors further develop their theory of approximate operations.…”