On color-automorphism vertex transitivity of semigroups

In this paper, we continue the study of Kelarev and Praeger devoted to the color-automorphism vertex transitivity of Cayley graphs of semigroups and we generalize and complete some of their results. For this purpose, first we show that for a semigroup S and a non-empty subset C ⊆ S , the C o l A u t C ( S ) -vertex-transitivity of C a y ( S , C ) is equivalent to the C o l A u t 〈 C 〉 ( S ) -vertex transitivity of C a y ( S , 〈 C 〉 ) , where 〈 C 〉 denotes the subsemigroup generated by C in S . Then we use this result to characterize a color-automorphism vertex transitive Cayley graph C a y ( S , C ) , where for every a ∈ S , 〈 C 〉 a is a simple 〈 C 〉 -act or for every a ∈ S , 〈 C 〉 a is finite. Similarly, we characterize a C o l A u t C ( S ) -vertex-transitive C a y ( S , C ) when for every c ∈ C , | 〈 c 〉 | is infinite and c is left cancellable. Finally, we use these results to establish that if S = ∪ ̇ α ∈ Y S α is a semilattice of semigroups S α and C is a non-empty subset of S , then the C o l A u t C ( S ) -vertex-transitivity of C a y ( S , C ) implies that Y has an identity e and C = C e . This answers an open question asked in a previous article.