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Groups and actions in transformation semigroups. (English) Zbl 0902.20028
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.

##### MSC:
 20M20 Semigroups of transformations, relations, partitions, etc. 20M10 General structure theory for semigroups 20B40 Computational methods (permutation groups) (MSC2010) 20F05 Generators, relations, and presentations of groups
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