A finiteness condition for finitely generated semigroups (Q5903841)

Let S be a finitely generated semigroup all of whose subgroups are locally finite. It is shown that, if for every \(s\in S\) there exists \(n\geq 2\) such that \(s=s^ n\) (that is S is a union of locally finite groups), then S is finite. This extends a result of \textit{A. de Luca} and \textit{A. Restivo} [Semigroup Forum 28, 123-134 (1984; Zbl 0529.20044)], and allows to derive as a consequence the theorem of Green and Rees on the relation between the restricted Burnside problems for groups and semigroups.