Power graphs and semigroups of matrices. (Q2721601)

By the power graph of a semigroup \(S\) the authors mean the directed graph with the set of vertices \(S\) and with edges \((u,v)\) where \(u,v\in S\) and \(v\) is a power of \(u\) but \(v\neq u\). Infinite groups whose power graphs satisfy a certain technical finiteness condition were characterized by the first two authors [in Contributions to general algebra 12. Klagenfurt, Verlag Johannes Heyn, 229-235 (2000; Zbl 0966.05040)]. In this note the result is extended to the case where \(S\) is an infinite semigroup of \(n\times n\) matrices over a division ring or a semigroup of \(n\times n\) monomial matrices over a group.