Representation of congruences on regular semigroups with inverse transversals (Q1806293)

Let \(S\) be a regular semigroup. An inverse subsemigroup \(S^0\) of \(S\) containing a unique inverse of each element of \(S\) is called an inverse transversal of \(S\) (this concept was introduced by \textit{T. S. Blyth} and \textit{R. McFadden} [Proc. R. Soc. Edinb., Sect. A 92, 253-270 (1982; Zbl 0507.20026)]). Based on a representation theorem of regular semigroups having an inverse transversal due to \textit{T. Saito} [Proc. Edinb. Math. Soc., II. Ser. 32, No. 1, 41-51 (1989; Zbl 0647.20063)] all congruences on \(S\) are described by means of congruences on the structural components of this representation. This process is simplified by a modification of the above representation provided by \textit{X. Tang} [Semigroup Forum 55, No. 1, 24-32 (1997; Zbl 0897.20040)]. Since any homomorphic image \(T\) of a regular semigroup \(S\) with inverse transversal also has an inverse transversal, the structure of \(T\) is described in terms of the components in the representation of \(S\).