On the Cayley semigroup of a finite aperiodic semigroup

TL;DR

This paper investigates the Cayley semigroup generated by functions derived from a finite aperiodic semigroup, proving that it is itself finite and aperiodic, thus extending understanding of semigroup structures.

Contribution

The paper introduces and analyzes the Cayley semigroup of a finite aperiodic semigroup, establishing its finiteness and aperiodicity, which was previously unconfirmed.

Findings

The Cayley semigroup generated by these functions is finite.

The Cayley semigroup is aperiodic.

The results extend the understanding of semigroup structures.

Abstract

Let $S$ be a finite semigroup. In this paper we introduce the functions $\phi_s:S^* \to S^*$, first defined by Rhodes, given by $\phi_s([a_1,a_2 ,...,a_n]) = [sa_1,sa_1a_2,..., sa_1a_2 ... a_n]$. We show that if $S$ is a finite aperiodic semigroup, then the semigroup generated by the functions $\{\phi_s\}_{s \in S}$ is finite and aperiodic.