Locally uniformly \(\pi\)-regular semigroups (Q2709040)

A semigroup \(S\) is \(\pi\)-regular (elsewhere in the literature eventually regular) if each \(a\) in \(S\) has a power that is regular. We say that \(S\) is completely \(\pi\)-regular (or is an epigroup, or is groupbound) if each element \(a\) of \(S\) has a completely regular power, that is a power that lies in some subgroup of \(S\). Finally, a \(\pi\)-regular semigroup whose regular elements are completely regular is called uniformly \(\pi\)-regular.NEWLINENEWLINENEWLINEA semigroup \(S\) is locally \(P\) is all of its local submonoids, \(eSe\) (\(e\) idempotent), have property \(P\). The main result of this paper gives five equivalent definitions of the property of local uniform \(\pi\)-regularity in terms of the group-bound and regular elements of the local submonoids.NEWLINENEWLINEFor the entire collection see [Zbl 0954.00028].