Abstract. In [1] we proved that a semigroup is a right commutative T1 semigroup if and only if it is isomorphic to a full $\Delta $-overact of a null semigroup by a commutative nil $\Delta $-semigroup with an identity adjoined (see Theorem 2.1 of [1]). Such overacts are defined by means of full $\Delta $-acts (see Construction 2.1 of [1]), and these acts are examined in the third section of [1] and are shown to be obtained by Construction 3.1 of [1]. In this note, we prove a simpler version of Theorem 2.1 of [1]: a semigroup is a right commutative T1 semigroup if and only if it is isomorphic either to a commutative nil $\Delta $-semigroup with an identity adjoined or to a semigroup $S=\{0,e,a\} $ obtained by adjoining to a null semigroup $A=\{0,a\} $ an idempotent $e$ that is both a right identity element of $S$ and a left annihilator element for $A$. This new fact shows that the constructions mentioned above provide only two very special types of semigroups and, actually, we do not need the notion of a full $\Delta $-overact and a full $\Delta $-act in order to describe right commutative T1 semigroups.

AMS Subject Classification (1991): 20M35

Received February 20, 2004, and in revised form September 16, 2004. (Registered under 5855/2009.)