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

Groups and Actions in Finite Transformation Semigroups.

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

Abstract.

Let

S

be a transformation semigroup of degree

n

. To each element

s

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

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

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

G

_L(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

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.

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