On weakly \(P\)-Ehresmann semigroups
This paper continues a tradition of exploring various ways of generalizing inverse semigroups. On the one hand, replacing the Green relations by generalized Green relations leads to the notions of restriction and, more generally, Ehresmann semigroups, along with broader classes still. On the other hand, within the class of regular semigroups, those having an inverse transversal and the regular \(^*\)-semigroups (those endowed with an involutory von Neumann inverse) evidence two different ways of narrowing the general class; those having a regular \(^*\)-transversal, usually required to be a quasi-ideal, clearly encompass both the latter. Now the reviewer [J. Pure Appl. Algebra 216, No. 3, 618--632 (2012; Zbl 1257.20058)] introduced \(P\)-restriction (\(P\)-Ehresmann) semigroups as common generalizations of restriction (Ehresmann) and regular \(^*\)-semigroups. Finally getting to the paper under review, the authors introduce the semigroups \((S, \cdot, ^+, ^*)\) of the title, which encompass both \(P\)-Ehresmann semigroups and regular semigroups having a regular \(^*\)-transversal that is a quasi-ideal. (To avoid conflict, the authors must use \(^\circ\) rather than \(^*\) for the operation in the transversal throughout.) The definition is by identities, which allows easy identification of various natural subclasses. One particular such subclass is then identified with the pseudo-Ehresmann semigroups studied by the first author [J. Aust. Math. Soc. 105, No. 2, 257--288 (2018; Zbl 1400.20063)], which encompassed the \(P\)-Ehresmann semigroups and the regular semigroups having an inverse transversal.\par This is not simply empty generalization. The authors provide a structure theorem for the semigroups of the title in terms of \(P\)-Ehresmann semigroups, their sets of projections and certain left and right normal partial bands, in the tradition of the original work on regular semigroups with inverse transversals by \textit{T. S. Blyth} and \textit{R. McFadden} [Proc. R. Soc. Edinb., Sect. A, Math. 92, 253--270 (1982; Zbl 0507.20026)].
scientific article; zbMATH DE number 7364723
scientific article