S. A. Linton, G. Pfeiffer, E. F. Robertson and N. Ruskuc,
Groups and Actions in 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
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.
Available as DVI file (67 kB) and as
compressed PostScript file (117 kB).