2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/167707Let $S$ be a finite semigroup. In this paper we introduce the functions $ϕ_s:S^* \to S^*$, first defined by Rhodes, given by $ϕ_s([a_1,a_2 ,...,a_n]) = [sa_1,sa_1a_2,..., sa_1a_2 ... a_n]$. We show that if $S$ is a finite aperiodic semigroup, then the semigroup generated by the functions $\{ϕ_s\}_{s \in S}$ is finite and aperiodic.Group Theorytext