On the \((m,r)\)-potent ranks of certain semigroups of transformations. (Q2788763)

An element \(x\) of a semigroup \(S\) is called \((m,r)\)-\textit{potent} if \(x^{m+r}=x^m\) and \(x,x^2,\ldots,x^{m+r-1}\) are all different. The main results in the paper under review assert that, for \(2\leq m+r\leq n+1\), the minimal cardinality of a generating set consisting only of \((m,r)\)-potent elements for the singular part of the full transformation monoid is \(n(n-1)/2\) and that the minimal cardinality of a generating set consisting only of \((m,r)\)-potent elements for the singular part of the full partial transformation monoid is \(n(n+1)/2\).