Groups and actions in transformation semigroups

Abstract.

Let $S$ be a transformation semigroup of degree $n$. To each element $s\in S$ 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 $\mathcal{R}$-class of $s$ is a disjoint union of certain sets, each having size equal to the size of $G_R(s)$. As a consequence, we show 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 we associate to $s$ another permutation group $G_L(s)$ on the image of $s$, and prove analogous results for the $\mathcal{L}$-class of $S$. Finally we prove 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.