# The endomorphism monoids and automorphism groups of Cayley graphs of semigroups

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 $$\mathrm {Cay}(S,C)$$ 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 $$\mathrm {Cay}(S,C)$$ Cay ( S , C ) [Proposition 2.3 and Corollary 2.4]. Then we use these results to characterize when $$\mathrm {Cay}(S,C)$$ Cay ( S , C ) is color-endomorphism vertex-transitive [Theorem 3.4]. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Semigroup Forum Springer Journals

# The endomorphism monoids and automorphism groups of Cayley graphs of semigroups

, Volume 95 (1) – May 15, 2017
13 pages

/lp/springer_journal/the-endomorphism-monoids-and-automorphism-groups-of-cayley-graphs-of-Vw7jHo0GM0
Publisher
Springer US
Subject
Mathematics; Algebra
ISSN
0037-1912
eISSN
1432-2137
D.O.I.
10.1007/s00233-017-9875-4
Publisher site
See Article on Publisher Site

### 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 $$\mathrm {Cay}(S,C)$$ 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 $$\mathrm {Cay}(S,C)$$ Cay ( S , C ) [Proposition 2.3 and Corollary 2.4]. Then we use these results to characterize when $$\mathrm {Cay}(S,C)$$ Cay ( S , C ) is color-endomorphism vertex-transitive [Theorem 3.4].

### Journal

Semigroup ForumSpringer Journals

Published: May 15, 2017

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