Abstract. A semigroup $S$ is said to have the Berman property if its $p_n$-sequence (counting the $n$-ary term operations which depend on all their variables) is either eventually strictly increasing or else it is bounded. In an earlier paper the authors showed that every surjective semigroup (i.e. a semigroup $S$ satisfying $S^2=S$) has the Berman property. In the present paper, we consider finite semigroups which are nilpotent ideal extensions of certain types of surjective semigroups and prove the Berman property for them. Our scope includes, for example, finite nilpotent extensions of completely regular semigroups (unions of groups) and of commutative semigroups. Also, the Berman property is proved for finite strict nilpotent extensions of arbitrary surjective semigroups.

AMS Subject Classification (1991): 20M10, 08A40

Received March 24, 1999. (Registered under 2718/2009.)