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S. A. Linton, G. Pfeiffer, E. F. Robertson, and N. Ruskuc,

Groups and Actions in Finite Transformation Semigroups.

Math. Z. 228 (1998), 435-450.


Let S be a transformation semigroup of degree n. To each element s in S we associate a permutation group GR(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 GR(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 GL(s) on the image of s, and prove analogous results for the L-class of s. Finally we prove that the Schützenberger group of the H-class of s is isomorphic to the intersection of GR(s) and GL(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.

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