Diagonal bi-acts over semigroups with finiteness conditions. (Q2634956)

It is proved that if the diagonal bi-act \(_S(S\times S)_S\) over an infinite epigroup \(S\) is finitely generated, then \(S=G\sqcup N\) (disjoint union), where \(G\) is an infinite group with finitely many conjugacy classes and \(N\) a certain ideal of \(S\). It is also proved that there does not exist an infinite locally finite group with a finite set of conjugacy classes. Therefore, as every locally finite semigroup is an epigroup, if the diagonal bi-act \(_S(S\times S)_S\) over a locally finite semigroup \(S\) is finitely generated, then \(S\) is finite.