Certain congruences on \(E\)-inversive \(E\)-semigroups
A semigroup \((S,\cdot)\) is called \(E\)-inversive if for every \(a\in S\), \(ax\in E(S)\) for some \(x\in S\) (where \(E(S)\) denotes the set of all idempotents of \(S\)) -- note that any regular or finite semigroup is \(E\)-inversive. \(S\) is called \(E\)-semigroup if \(E(S)\) forms a subsemigroup of \(S\). \(S\) is called \(E\)-dense if \(S\) is \(E\)-inversive and the idempotents of \(S\) commute. Let \(S\) be an \(E\)-inverse \(E\)-semigroup. In this paper, first a new characterization of the least group congruence on \(S\) is given by means of \(W(a)=\{x\in S\mid x=xax\}\) for \(a\in S\). Next, a semilattice congruence on \(S\) is defined which for eventually regular \(E\)-semigroups is the least such congruence. Furthermore, ``small'' regular congruences on \(S\) are found, which on certain regular semigroups are equal to the identity relation. It is shown that in general, for \(E\)-inversive \(E\)-semigroups (even for eventually regular \(E\)-semigroups) a least regular congruence does not exist. Finally, the idempotent separating congruence \(\mu\) defined by \textit{P. Edwards} for arbitrary semigroups [Bull. Aust. Math. Soc. 28, 23-38 (1983; Zbl 0511.20044)] is characterized for \(E\)-inversive \(E\)-semigroups. It is shown that in the case that \(E(S)\) forms a rectangular band, \(\mu\) is the greatest idempotent separating congruence. Also a characterization of \(\mu\) is given for \(E\)-dense semigroups.
scientific article; zbMATH DE number 1837421
scientific article