RESEARCH ARTICLE

Endomorphisms of Cayley digraphs of rectangular groups

Srichan Arworn^a, Boyko Gyurov^b, Nuttawoot Nupo^a, Sayan Panma<sup>c

a</sup> Department of Mathematics, Chiang Mai University, Huay Kaew Road, Chiang Mai 50200, Thailand
^b School of Science and Technology, Georgia Gwinnett College, University System of Georgia
^c Center of Excellence in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

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

Keywords: Cayley digraphs, rectangular groups, endomorphisms.

MSC: 05C20, 05C25, 20K30, 20M99

Received: 11.01.2017
Revised: 09.12.2018

Language: English