Abstract. Let $T$ be a locally $\cal R$-unipotent semigroup acting on a semigroup $S$ by endomorphisms on the left. A kind of semidirect product of $S$ by $T$ is defined, which leads to a regular (locally $\cal R$-unipotent) semigroup if $S$ is regular (locally $\cal R$-unipotent). On the level of e-varieties we naturally obtain a binary operation ``$*$'' which satisfies $({\cal U}*{\cal V})*{\cal W}={\cal U}*({\cal V}*{\cal W})$ whenever $\cal U$ is an e-variety, $\cal V$ and $\cal W$ are e-varieties of locally $\cal R$-unipotent semigroups and ${\cal U}*({\cal V}*{\cal W})$ is generated by $\{A*(B*C) | A\in{\cal U}, B\in{\cal V}, C\in{\cal W}\} $. This generalizes recent results found in [1], [4], [9].

AMS Subject Classification (1991): 20M10, 20M17

Received March 25, 1998, and in final form August 4, 2000. (Registered under 2778/2009.)