An order theoretic characterization of locally orthodox regular semigroups (Q1117332)

A partial order on a regular semigroup \(S\) is introduced by defining \(x\leq_ ey\) iff \({\mathcal R}_ x\leq {\mathcal R}_ y\), \({\mathcal L}_ x\leq {\mathcal L}_ y\) and there exists idempotents \(e, f\) such that \(e{\mathcal R}x\), \(f{\mathcal L}x\) and \(x=eyf\). The relation \(\leq_ e\) contains the natural partial order on a regular semigroup \(S\) whereby \(x\leq y\) iff there exist idempotents \(e, f\) such that \(x=ey=yf\). The author shows that the two partial orders agree iff \(S\) is locally orthodox. This result can be compared with the theorem of Nambooripad that the natural partial order is compatible with multiplication iff \(S\) is locally inverse and the more general result of \textit{T. S. Blyth} and \textit{G. Gomes} [Proc. Edinb. Math. Soc., Sect. A 94, 79-84 (1983; Zbl 0511.20043)] that \(\leq\) is right compatible iff \(S\) is a locally L-unipotent (and thus in particular a locally orthodox) semigroup.