A combinatorial property and Cayley graphs of semigroups
type: Item
instance of: scholarly article
title: A combinatorial property and Cayley graphs of semigroups
zbMATH Open document ID: 1016.20047
published in: Semigroup Forum
publication date: 2003-03-18T00:00:00Z
review text: Analogously to the Cayley graphs of groups, the Cayley graphs of semigroups are studied here. Let \(S\) be a semigroup and let \(T\) be a subset. The Cayley graph \(\text{Cay}(S,T)\) is the directed graph whose vertex set is \(S\) and in which an edge goes from \(u\) to \(v\) if and only if \(u\neq v\) and \(v=ut\) for some \(t\in T\). Now let \(D\) be a finite digraph. A semigroup \(S\) is said to be Cayley \(D\)-saturated with respect to a subset \(T\) of \(S\), if for each infinite subset \(V\) of \(T\) there exists a subgraph of \(\text{Cay}(S,T)\) isomorphic to \(D\). The main results of the paper are the following two theorems. Theorem 2.1. Let \(D\) be a finite graph which is not null, and let \(S\) be an infinite semigroup. Then the following conditions are equivalent: (i) \(S\) is Cayley \(D\)-saturated with respect to \(S\); (ii) \(S\) does not have an infinite set of pairwise incomparable principal right ideals, and if \(D\) has a cycle, then the number of principal right ideals in \(S\) is finite. Theorem 2.2. Let \(D\) be a finite graph which is not null. Each infinite semigroup \(S\) has an infinite subset \(T\) such that \(S\) is not \(D\)-saturated with respect to \(T\).
reviewed by: Bohdan Zelinka
Mathematics Subject Classification ID: 20M30
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 05C20
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20M12
zbMATH DE Number: 1883941
zbMATH Keywords: Cayley graphs of semigroups
zbMATH Keywords: directed graphs
zbMATH Keywords: finite digraphs
zbMATH Keywords: Cayley \(D\)-saturated semigroups
zbMATH Keywords: principal right ideals
author: A. V. Kelarev
author: Stephen J. Quinn
MaRDI profile type: Publication
full work available at URL: s002330010162
OpenAlex ID: W2003865102
DOI: 10.1007/S002330010162
description: scientific article; zbMATH DE number 1883941