Groups and Actions in Transformation Semigroups

Let S be a transformation semigroup of degree n. To each element s is an element of S we associate a permutation group G(R)(S) acting on the image of s, and we find a natural generating set for this group. It turns out that the 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, we show that two R-classes containing elements with equal images have the same size, even if they do not belong to the same D-class. By a certain duality process we associate to s another permutation group G(L)(s) on the image of s, and prove analogous results for the L-class of S. Finally we prove that the Schutzenberger group of the 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.