Bisimple monogenic orthodox semigroups. (Q1014264)

Let \(S\) be an orthodox semigroup generated by two elements \(a\) and \(b\) such that \(aba=a\) and \(bab=b\). Such a semigroup is called a monogenic orthodox semigroup by analogy with the inverse semigroup case and is denoted by \(S=\langle a,b\rangle\). The main result is a complete description of the semigroup \(S=\langle a,b\rangle\) in the case it is bisimple and \(a\) (and hence \(b\)) is a nongroup element of \(S\) (this means that \(a\) does not belong to any subgroup of \(S\)). Separately, the case is considered when \(a\) (and hence \(b\)) is a group element (in this case \(S\) is necessarily bisimple).