ISSN:
0025-5874
Keywords:
Mathematics Subject Classification (1991):20M20, 20B40, 20M10
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/PL00004628
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