Endomorphisms of Cayley digraphs of rectangular groups

Abstract

Let $\mathrm{Cay}(S,A)$ denote the Cayley digraph of the semigroup $S$ with respect to the set $A$, where $A$ is any subset of $S$. The function $f : \mathrm{Cay}(S,A) \to \mathrm{Cay}(S,A)$ is called an endomorphism of $\mathrm{Cay}(S,A)$ if for each $(x,y) \in E(\mathrm{Cay}(S,A))$ implies $(f(x),f(y)) \in E(\mathrm{Cay}(S,A))$ as well, where $E(\mathrm{Cay}(S,A))$ is an arc set of $\mathrm{Cay}(S,A)$. We characterize the endomorphisms of Cayley digraphs of rectangular groups $G\times L\times R$, where the connection sets are in the form of $A=K\times P\times T$.