Group congruences on an \(E\)-inversive semigroup (Q1369329)

A semigroup \(S\) is called \(E\)-inversive if for every \(a\in S\) there exists \(x\in S\) such that \(ax\) is an idempotent of \(S\). In this paper group congruences on \(S\) are studied using their description by means of the members of the set \(C\) of all full, selfconjugate and unitary subsemigroups of \(S\) [the reviewer, J. Aust. Math. Soc., Ser. A 48, No. 1, 66-78 (1990; Zbl 0691.20050)]. The following is proved: 1) if \(H\in C\) and \(\rho_H\) is the corresponding group congruence then \(\text{Ker }\rho_H=H\); 2) if \(\gamma\) is a group congruence then \(\text{Ker }\gamma\in C\) and \(\gamma=\rho_{\text{Ker }\gamma}\); 3) for every congruence \(\rho\) and every group congruence \(\gamma\), \(\rho\vee\gamma=\gamma\circ\rho\circ\gamma\). Several characterizations of group congruences are given and their classes are described. Also, the join \(\rho\vee\gamma\) and its kernel are investigated. Finally, for \(E\)-inverse, \(E\)-unitary subgroups \(S\) it is shown that the mapping \(\Phi\colon\rho\to\sigma\vee\rho\) (where \(\sigma\) denotes the least group congruence on \(S\)) is a homomorphism of the congruence lattice of \(S\) onto the lattice of all group congruences on \(S\). Remark: Using the natural partial order on \(S\) similar results were obtained by \textit{S. Reither} in her Ph.D.-thesis (Univ. Vienna, 1995).