Regular semigroups with inverse transversals
Let \(S\) be a regular semigroup with set of idempotents \(E(S)\). An inverse subsemigroup \(T\) of \(S\) is said to be an inverse transversal of \(S\) if \(T\) contains a unique inverse \(x^0\) for each \(x\in S\). If \(S\) is a regular semigroup with inverse transversal, it is shown that the subsets \(I=\{e\in E(S)\mid e=ee^0\}\) and \(\Lambda=\{f\in E(S)\mid f=f^0f\}\) are subsemigroups of \(S\), simplifying the structure theorem of \textit{T. Saito} [Proc. Edinb. Math. Soc., II. Ser. 32, No. 1, 41-51 (1989; Zbl 0647.20063)]. The author also gives necessary and sufficient conditions for an inverse subsemigroup \(T\) of \(S\) to be an inverse transversal of \(S\), and specialized conditions when \(S\) is \(E\)-solid or locally inverse.
scientific article; zbMATH DE number 1045577