The maximal subsemigroups of the singular part of endomorphism monoids of the star graphs

Abstract

In this paper, we determine the maximal subsemigroups of the singular part of endomorphism monoids of the star graphs $S_n$, for a given positive integer $n$, namely $End(S_n), wEnd(S_n)$, and $swEnd(S_n)$, respectively. The monoid $\wEnd(S_n)$ of all weak endomorphisms on $S_n$, the monoid $\End(S_n)$ of all endomorphisms on $S_n$ as well as the monoid $\swEnd(S_n)$ of all strong weak endomorphisms on $S_n$ is regular. We also determine the maximal regular submonoids of the singular part of $\wEnd(S_n), \End(S_n)$, and $\swEnd(S_n)$.