Groups and actions in transformation semigroups (Q1266188)

Let \(S\) be a transformation semigroup of degree \(n\). To each element \(s\in S\) a permutation group \(G_R(s)\), acting on the image of \(s\), is associated. A natural generating set for this group is found. It turns out that the \(\mathcal R\)-class of \(s\) is a disjoint union of certain sets, each having size equal to the size of \(G_R(s)\). As a consequence, it is shown that two \(\mathcal R\)-classes containing elements with equal images have the same size, even if they do not belong to the same \(\mathcal D\)-class. By a certain duality process one associates to \(s\) another permutation group \(G_L(s)\) on the image of \(s\), and proves analogous results for the \(\mathcal L\)-class of \(S\). Finally, it is proved that the Schützenberger group of the \(\mathcal H\)-class of \(s\) is isomorphic to the intersection of \(G_R(s)\) and \(G_L(s)\). The results of this paper can also be applied in new algorithms for investigating transformation semigroups, which will be described in a forthcoming paper.