Finite semigroups with slowly growing pn-sequences

The p _n -sequence of a semigroup S is said to be polynomially bounded, if there exist a positive constant c and a positive integer r such that the inequality p _n (S) ≤cn ^r holds for all n≥ 1. In this paper, we fully describe all finite semigroups having polynomially bounded p _n -sequences. First we give a characterization in terms of identities satisfied by these semigroups. In the sequel, this result will allow an insight into the structure of such semigroups. We are going to deal with certain ideals and the construction of ideal extension of semigroups. In addition, we supply an effective procedure for deciding whether a finite semigroup has polynomially bounded p _n -sequence and give some examples. Received March 5, 1999; accepted in final form November 1, 1999.