The identity of completely \(\pi\)-regular semigroup rings (Q2745473)

Fifteen years ago, \textit{J. S. Ponizovskij} suggested the question [in Semigroup Forum 36, 1-46 (1987; Zbl 0629.20039)]: under what conditions does a semigroup algebra possess an identity? This is an interesting question and has been investigated by some mathematicians [see \textit{F. Li}, Semigroup Forum 46, No. 1, 27-31 (1993; Zbl 0787.16024)].NEWLINENEWLINENEWLINEIn this paper, the author investigates the existence of an identity of a completely \(\pi\)-regular semigroup. A semigroup is called completely \(\pi\)-regular if \(\forall a\in S\), \(\exists n\in\mathbb{N}\) such that \(a^n\) is an element of a group. The major result in this paper is: Suppose that \(S\) is a completely \(\pi\)-regular semigroup, then \(RS\) possesses an identity iff so does \(R\langle E(S)\rangle\) and there exists a finite subset \(U\subseteq E(S)\) such that \(S=SU=US\). Meanwhile, he obtained a characterization on completely [0-]simple semigroups, that is, a [0-]simple semigroup is completely simple iff \(S\) is left \(\pi\)-regular and possesses a non-zero idempotent.