A partial order in completely regular semigroups
Let S be a semigroup and let \({\mathcal S}\) be the relation defined on S by a\({\mathcal S}b\) if \(a^ 2=ab=ba\). S is called quasi-separative if for a,b\(\in S\) the equalities \(a^ 2=ab=ba=b^ 2\) hold only if \(a=b\). The author proves that completely regular semigroups are quasi-separative (S is completely regular if \(a\in (Sa^ 2)\cap (a^ 2S)\) for each \(a\in S)\). The converse is also true for the wider class of strongly \(\pi\)- regular semigroups (those satisfying for each \(a\in S\) the property that there exist positive integers m, n such that \(a^ m\in a^{m+1}S\) and \(a^ n\in Sa^{n+1})\). Whenever S is a completely regular semigroup, the relation \({\mathcal S}\) is a partial order on the set S. Also studied are connections between \({\mathcal S}\) and other relations which become partial orders on S when S has certain prescribed properties which are close to quasi-separativity.
scientific article; zbMATH DE number 3925125
scientific article