On the Cayley $\mathcal {D}$-saturated property of semigroups

Abstract

Recently, the Cayley $\mathcal {D}$-saturated property of semigroups has been introduced as a combinatorial property of semigroups. In this paper, we study the Cayley $\mathcal {D}$-saturated property of product semigroups and semilattice of semigroups. Let $S_1$ and $S_2$ be semigroups and $S=S_1\times S_2$. We denote by $\pi _i$, the canonical projection onto the $i$th component. We show that the Cayley $\mathcal {D}$-saturated property of $S$ with respect to $C\subseteq S$, is closely related to the Cayley $\mathcal {D}$-saturated property of $S_1$ with respect to $\pi _1(C)$ and the Cayley $\mathcal {D}$-saturated property of $S_2$ with respect to $\pi _2(C)$. Then we prove that for a semilattice of semigroups $S=\dot{\cup }_{\alpha \in Y}S_\alpha$, the Cayley $\mathcal {D}$-saturated property of $S$ with respect to $C\subseteq S$, is closely related to the Cayley $\mathcal {D}$-saturated property of $Y$ with respect to $X=\{x\in Y|S_x\cap C\ne \emptyset \}$. As consequences of our results, we characterize Cayley $\mathcal {D}$-saturated semigroup $S$ with respect to $C\subseteq S$, when $S$ is a group, left (right) group, rectangular band, rectangular group, completely simple semigroup, completely regular semigroup and Clifford semigroup. Also as a part of our work, we show that a graph $\Gamma$ is a Cayley graph of some rectangular band if and only if there exist some sets $A$ and $B$ such that $\Gamma$ is the disjoint union of graphs $\{\Gamma _i\}_{i\in I}$ such that for every $i\in I, \Gamma _i=(|B| K_1)+\overrightarrow{K_{|A|}}$. Then we use this characterization to generalize the previous results about rectangular bands and rectangular groups.