Semigroup Forum (2017) 95:179–191
The endomorphism monoids and automorphism groups
of Cayley graphs of semigroups
Received: 23 September 2016 / Accepted: 23 April 2017 / Published online: 15 May 2017
© Springer Science+Business Media New York 2017
Abstract 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 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 Cay(S, C) [Proposition 2.3 and Corollary 2.4].
Then we use these results to characterize when Cay(S, C) is color-endomorphism
vertex-transitive [Theorem 3.4].
Keywords Cayley graphs of semigroups · Vertex-transitive graphs · The automor-
phism groups of Cayley graphs of semigroups · The endomorphism monoids of
Cayley graphs of semigroups
1 Introduction and preliminaries
Recall that a graph = (V, E) is a set V = V () of vertices, together with a binary
relation E = E() on V . The elements of E are called the arcs of . Throughout this
paper, by a graph we mean a directed graph without multiple arcs, but possibly with
loops. Let S be a semigroup and C be a non-empty subset of S.TheCayley graph
Cay(S, C) of S with respect to C is deﬁned as the graph with vertex set S and arc
Communicated by Jorge Almeida.
Department of Mathematics, Institute for Advanced Studies in Basic Sciences, Zanjan