On the structure of glrac semigroups

Glrac semigroups were introduced by Branco, Gomes and Gould as a way of generalizing the notion of left restriction semigroup to similar structures whose projections form a left regular band, rather than a semilattice, while retaining in particular the left congruence and left ample conditions (whence the initialism). While there has been considerable attention paid to these semigroups recently, it is the author's contention that they are best studied within the framework of the increasingly broad classes of unary semigroups (S,& BULL;+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(S, \cdot \, <^>+)$$\end{document} that have since been studied intensively, from the left P-Ehresmann semigroups of the author onward to the D-semigroups of Stokes and even beyond. This article justifies the claim by first showing that glrac semigroups are precisely the left P-Ehresmann semigroups that satisfy the left ample condition. Then the author's representation theorem for left P-Ehresmann semigroups is used to show that glrac semigroups are, up to isomorphism, simply the spined products of (unary) semigroups of endomorphisms of left regular bands with left restriction semigroups, a theorem that can be tightly refined. Moreover, a weaker version of the left ample condition yields an analogous theorem for the even more ungainly named class of glrwac semigroups, where left Ehresmann semigroups replace left restriction semigroups. As an illustration of its utility, an application to proper covers is outlined.