Complete lattice homomorphism of strongly regular congruences on \(E\)-inversive semigroups. (Q2800021)

A semigroup \(S\) is \(E\)-inversive if for any \(a\in S\), there exists \(x\in S\) such that \(ax\in E(S)\), the set of idempotents of \(S\); equivalently if for each \(a\), there is a `weak inverse' \(x\), such that \(x=xax\). A congruence \(\rho\) on such a semigroup is called strongly regular if each \(a\in S\) has a weak inverse \(a'\) such that \(a\rho aa'a\). Although in that case \(S/\rho\) is regular, the converse need not be true.NEWLINENEWLINE The authors extend many of the main results of \textit{F. Pastijn} and \textit{M. Petrich} [Trans. Am. Math. Soc. 295, 607-633 (1986; Zbl 0599.20095)], on congruences on regular semigroups. For instance, the trace and kernel of a strongly regular congruence \(\rho\) on \(S\) are again defined as the restriction of \(\rho\) to \(E(S)\) and as the set \(E(S)\rho\), respectively. Left and right traces are also defined. The triple consisting of the kernel and left and right traces is then characterized and an isomorphism established between the lattice of strongly regular congruences and the appropriate lattice of such triples. The authors prove that the mapping from strongly regular congruences to their traces is a complete lattice homomorphism, thus generalizing the theorem of \textit{P. G. Trotter} [Semigroup Forum 34, 245-252 (1986; Zbl 0606.20053)], which solved the problem posed in the cited paper of Pastijn and Petrich. (The authors term this problem `open' throughout but, once answered, it is of course no longer `open'.)