Inverse transversals -- a guided tour (Q2709034)

Let \(S\) be a regular semigroup. An inverse subsemigroup of \(S\), which contains precisely one inverse \(x^\circ\) of every \(x\in S\) is called an inverse transversal of \(S\), denoted by \(S^\circ\). This concept has its origin in a paper by D. B. McAlister-T. S. Blyth. Its theory was developed in about 30 papers up to now. The paper under review gives a survey on all the known results adding some new ones.NEWLINENEWLINENEWLINEIn Section 1 basic properties of inverse transversals \(S^\circ\) are listed (including their proofs). Section 2 gives a classification of several types of inverse transversals, and their impact on the structure of \(S\) is described. Section 3 contains examples of regular semigroups \(S\) with particular inverse transversals \(S^\circ\). Section 4 contains the result that inverse transversals are mutually isomorphic, a construction of all regular semigroups with a quasi-ideal inverse transversal, and the result by T. Saito that a regular semigroup \(S\) is completely simple if and only if \(S^\circ\) is a group. In section 5 results on congruences on \(S\) with respect to \(S^\circ\) are collected. In particular, it contains a construction of all congruences by means of balanced linked triples (which depend on the congruences on \(S^\circ\)), and solutions of problems concerning the congruence extension property of \(S^\circ\).NEWLINENEWLINEFor the entire collection see [Zbl 0954.00028].