Quasi-orthodox semigroups with inverse transversals
Let S be a regular semigroup. A subsemigroup \(S\circ\) of S is called an inverse transversal of S if \(S\circ\) contains a single inverse of every element of S. Regular semigroups with inverse transversals were first studied by Blyth and McFadden (1982). In case that an inverse transversal \(S\circ\) of a regular semigroup S is a quasi-ideal of S, a complete description of the structure of S was given by McAlister and McFadden (1983). A regular semigroup S is called quasi-orthodox if there exist an inverse semigroup \(\Gamma\) and a surjective homomorphism \(\phi\) : \(S\to \Gamma\) such that \(\lambda \phi^{-1}\) is a completely simple subsemigroup of S for each idempotent \(\lambda\) of \(\Gamma\). In this paper, we first obtain a necessary and sufficient condition in order for regular semigroups with inverse transversals to be quasi- orthodox. The inverse transversal \(S\circ\) of a quasi-orthodox semigroup S does not necessarily need to be a quasi-ideal of S. Let S be a regular semigroup with an inverse transversal \(S\circ\). If \(x\in S\), the inverse of x which belongs to \(S\circ\) is denoted by \(x\circ\). The sets \(I=\{e\in S:\) \(ee\circ =e\}\) and \(\Lambda =\{f\in S:\) \(f\circ f=f\}\) play an important role to investigate the structure of S. McAlister and McFadden showed that, if \(S\circ\) is a quasi-ideal of S, then they are subbands of S. We here show that I and \(\Lambda\) are subbands of S when S is a quasi- orthodox semigroup. In case that I and \(\Lambda\) are subbands of S, a structure theorem of a regular semigroup S with an inverse transversal \(S\circ\) was given by the author (1984). By using the result, we obtain a structure theorem of quasi-orthodox semigroups with inverse transversals.
scientific article; zbMATH DE number 4002389