Power Powers And Generalized Cayley Graphs

In this paper, we study idempotents and potents of (n, m)-semigroups and gener-alized Cayley graphs of a class of semigroups.First, we introduce the definition of the power of an (n, m)-semigroup, then givethe concept of the idempotents and potents of an (n, m)-semigroup, establish the corre-sponding exponential law, and describe the relation between idempotents and potents.We then give some necessary and sufcient conditions for an element of an (n, m)-groupto be idempotent and potent, respectively and also prove that the set of all potents ofstrongly reversible E-(n, m)-semigroup is an ideal. All these generalize the correspond-ing notions and results for usual binary and n-ary semigroups.Second, for a class of semigroups S which satisfies the condition x2y=xy=xy2(x, y∈S), we describe their generalized Cayley graphs, and further give the gener-alized Cayley graphs of a0-direct union of such semigroups. What’s more, we describethe generalized Cayley graphs of classes of semigroups satisfying one of the above twoequations.