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In this paper, we define a congruence   $\eta ^{\ast }$   onsemigroups. For the finite semigroups   $S$  ,$\eta ^{\ast}$   is the smallest congruence relation suchthat   $S/\eta ^{\ast}$   is a nilpotent semigroup (in the sense ofMal'cev). In order to study the congruence relation   $\eta ^{\ast}$   on finite semigroups, we define a$\textbf{CS}$  -diagonal finite regular Rees matrixsemigroup. We prove that if   $S$   is a$\textbf{CS}$  -diagonal finite regular Rees matrixsemigroup, then   $S/\eta ^{\ast }$   is inverse. Also, if$S$   is a completely regular finitesemigroup, then   $S/\eta ^{\ast }$   is a Cliffordsemigroup.

THE CONGRUENCE η* ON SEMIGROUPS