Extensions of algebraic systems

There are several generalizations to universal algebras of the notion "The groupA\mathfrak {A}is an extension of the groupB\mathfrak {B}by the groupC\mathfrak {C}". In this paper we study three such generalizations and the corresponding products of classes of algebraic systems. Various results are presented. One such theorem characterizes the weakly congruence regular varieties admitting extensions of a particular sort. Another result gives, under a weak congruence permutability condition, an equational basis for the variety obtained by applying one such product to two other varieties.