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- Title
On the Cayley $$\mathcal {D}$$ -saturated property of semigroups.
- Authors
Khosravi, Behnam; Khosravi, Behrooz; Khosravi, Bahman
- 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.
- Subjects
CAYLEY algebras; SEMIGROUPS (Algebra); SEMILATTICES; CLIFFORD algebras; CAYLEY graphs
- Publication
Semigroup Forum, 2015, Vol 91, Issue 2, p502
- ISSN
0037-1912
- Publication type
Article
- DOI
10.1007/s00233-015-9716-2