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Let S be a finite semigroup. In this paper we introduce the func- tions ϕs : S∗ → S∗, first defined by Rhodes, given by ϕs((a1, a2, . . . , an)) = (sa1, sa1a2, . . . , sa1a2    an). We show that if S is a finite aperiodic semigroup, then the semigroup generated by the functions {ϕs}s∈S is finite and aperiodic.

ON THE CAYLEY SEMIGROUP OF A FINITE APERIODIC SEMIGROUP