Semigroups that contain all singular transformations. (Q1434128)

Denote by \({\mathcal T}_N\) and \({\mathcal S}_N\) the semigroups of all transformations and all permutations respectively on the set \(X=\{1,2,\dots,N\}\). Let \(G\) be a group of permutations on \(X\) and let \(\mathcal U\) be a collection of transformations on \(X\) each with rank \(N-1\). The semigroup \(\langle G,{\mathcal U}\rangle\) is referred to as a near permutation semigroup on \(X\). In this paper, the author finds necessary and sufficient conditions in order that \({\mathcal T}_N\setminus{\mathcal S}_N\subseteq\langle G,{\mathcal U}\rangle\).