Finite semigroups with few term operations

For a semigroup S, p(n)(S) denotes the number of n-ary term operations of S depending on all their variables. The purpose of this paper is to study finite semigroups S with the property that their p(n)-sequence p(S) = <p0(S), p1(S),...> is bounded. Such semigroups are described first in terms of identities and then structurally as nilpotent extensions of semilattices, Boolean groups and rectangular bands. As a corollary it is shown that if p(S) is bounded then eventually either p(n)(S)= 0 or 1. It is also shown that there is an effective procedure which decides whether the p(n)-sequence of a given finite semigroup is bounded or not. (C) 2001 Elsevier Science B.V. All rights reserved.