The endomorphism monoids and automorphism groups of Cayley graphs of semigroups

In this note, we introduce the notions of color-permutable automorphisms and color-permutable vertex-transitive Cayley graphs of semigroups. As a main result, for a finite monoid S and a generating set C of S, we explicitly determine the color-permutable automorphism group of (mathrm {Cay}(S,C)) [Theorem 1.1]. Also for a monoid S and a generating set C of S, we explicitly determine the color-preserving automorphism group (endomorphism monoid) of (mathrm {Cay}(S,C)) [Proposition 2.3 and Corollary 2.4]. Then we use these results to characterize when (mathrm {Cay}(S,C)) is color-endomorphism vertex-transitive [Theorem 3.4].