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 be a transformation semigroup of degree .
To each element in we associate a permutation group
acting on the image of , and we find a natural
generating set for this group. It turns out that the -class
of is a disjoint union of certain sets, each having
size equal to the size of .
As a consequence, we show that two -classes containing elements
with equal images have the same size, even if they do not
belong to the same -class.
By a certain duality process we associate to
another permutation group on the image of ,
and prove analogous results for the -class of .
Finally we prove that the Schützenberger group of
the -class of is isomorphic to the intersection
of and . 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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