An Upper Bound for the Complexity of Transformation Semigroups

A transformation semigroup (ts) X = (Q,, S,) in this paper consists of a finite set of states Qx and a subsemigroup of transformations S, of PF(Q,), the monoid of all partial functions on Q, with composition as multiplication. For n > 1, ii’ denotes the transformation monoid (tm) with n states and with the n constant maps on those n states (along with the identity function) as transformations. The main theorem (Theorem 2.2) of this paper shows that if a ts X does not contain a copy of ii’ as a sub-ts, then the complexity of X is less than n. Recall that a ts X has complexity less than or equal to II (written Xc Q n) iff