Introduction to \(E\)-inversive semigroups (Q2709041)

This is a useful survey on \(E\)-inverse semigroups. Thierrin (1952) defined a semigroup \(S\) to be \(E\)-inverse if for every \(a\in S\) there exists \(x\in S\) such that \(ax\in E_S\) (the set of all idempotents of \(S\)). Examples are regular semigroups or periodic (in particular, finite) semigroups. Thierrin (1952, 1956) investigated \(E\)-inverse semigroups \(S\), which are rectangular (i.e. \(ax=by=az=m\) implies \(bz=m\) for \(a,b,x,y,z\in S\)) providing a construction of these semigroups. Further investigations of these semigroups are due to Clifford (1956), Yamada (1955), and Petrich (1963). The present article gives an up-to-date survey on \(E\)-inverse semigroups following the notation and terminology of the standard monographs of \textit{J. M. Howie} [Fundamentals of semigroup theory, Clarendon Press, Oxford (1995; Zbl 0835.20077)] and of \textit{M. Petrich} [Introduction to semigroups, Merill, Columbus (1973; Zbl 0321.20037)]. After a brief introduction containing basic definitions and examples, Section 2 is devoted to various characterizations, Section 3 contains cancellativity conditions, Section 4 discusses restrictions on idempotents and the final Sections 5 and 6 present results concerning congruences and covers.NEWLINENEWLINEFor the entire collection see [Zbl 0954.00028].