k-TRANSFORMATION SEMIGROUPS OF TILSON

Let Q be a finite set and let k be a non negative integer. A (partial) function f: Q + Q is a k-map if ]sf’] < k, Vq E Q. A transformation semigroupx = (Q, S) is a k-t.s. if each s ES is a k-map. Let TS be the collection of all (finite) transformation semigroups and let c: TS + N be the complexity function. For background on complexity see [2]. The main theorem of this paper shows that if X is a k-t.s., then Xc c k. This theorem is then used to prove the following conjecture of B. Tilson: Let S be a finite semigroup and let R be an $class of S. Define Rt to be the number of distinct idempotents of R and let St = max{Rt ) R is an 3 class of S}. Then SC < St. In the last section a necessary and sufficient condition is given in order that a semigroup be faithfully represented as a semigroup of k-maps. For undefined terms in this paper see [2].