Left semiregular orthodox semigroups and regular orthodox semigroups with inverse transversals. (Q1769499)

An inverse transversal of a regular semigroup \(S\) is an inverse subsemigroup of \(S\) which contains exactly one inverse \(x^\circ\) of each element \(x\in S\). In this paper, regular semigroups \(S\) with an inverse transversal are investigated, which satisfy either (i) \(E_S\) is a left semiregular band (i.e., \(efg=efgegfg\) for all \(e,f,g\in E_S\)), or (ii) \(E_S\) is a regular band (i.e., \(efge=efege\) for all \(e,f,g\in E_S\)). The main results provide constructions of such semigroups \(S\) by means of right inverse semigroups \(R\) with inverse transversal and semilattices \(I\) of left zero semigroups with a semilattice transversal. More precicely, it is shown, that \(S\) is isomorphic with \(I*R\), where \(I=\{e\in E_S\mid e=ee^\circ\}\), \(R=\{a\in S\mid a=a^{\circ\circ}a^\circ a\}\) and \(I*R\) is a certain subset of \(I\times R\) endowed with a multiplication similar to the semidirect product.