Abstract. A semigroup is called right commutative if it satisfies the identity $axy=ayx$. We say that a semigroup $S$ is a $\Delta $-semigroup if the lattice of all congruences of $S$ is a chain. In this paper we prove that a semigroup is a right commutative $\Delta $-semigroup if and only if it is isomorphic to either $G$ or $G^0$, where $G$ is a subgroup of a quasicyclic $p$-group ($p$ is a prime) or $L$ or $L^0$, where $L$ is a two-element left zero semigroup or $N$, where $N$ is a non-trivial right commutative nil semigroup whose lattice of ideals is a chain or a full $\Delta $-overact of a null semigroup by a commutative nil $\Delta $-semigroup with an identity adjoined. We also give a construction of full $\Delta $-overacts of null semigroups by commutative nil $\Delta $-semigroups with an identity adjoined.

AMS Subject Classification (1991): 20M35

Received March 4, 1998, and in final form September 29, 1999. (Registered under 2719/2009.)