Weak inverses on eventually regular semigroups. (Q1768173)

An element \(y\) of a semigroup \(S\) is said to be a weak inverse of \(x\in S\) if \(yxy=y\), and an inverse of \(x\) if also \(xyx=x\). Let \(W(x)\) denote the set of weak inverses of \(x\), and \(V(x)\) the set of inverses of \(x\). The authors obtain analogues of the result of \textit{T. E. Hall} [Bull. Aust. Math. Soc. 1, 195-208 (1969; Zbl 0172.31101)] that \(\gamma=\{(x,y)\in S\times S\mid V(x)=V(y)\}\) is the least inverse semigroup congruence on an orthodox semigroup. A semigroup \(S\) is said to be eventually regular if for each \(a\in S\), \(a^n\) is regular for some positive integer \(n\). An eventually regular semigroup is said to be eventually orthodox (resp. eventually inverse) if its idempotents form a subsemigroup (resp. semilattice). The authors note that \(\delta=\{(x,y)\in S\times S\mid W(x)=W(y)\}\) is a congruence on any semigroup \(S\), and they study \(\delta\) for semigroups \(S\) that are eventually regular. They show, for instance, that \(\delta\) is an eventually inverse semigroup congruence on an eventually orthodox semigroup.