TY - JOUR

T1 - Groups and Actions in Transformation Semigroups

AU - Linton, Stephen Alexander

AU - Pfeiffer, G

AU - Robertson, Edmund Frederick

AU - Ruskuc, Nikola

PY - 1998/7

Y1 - 1998/7

N2 - Let S be a transformation semigroup of degree n. To each element s is an element of 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 Schutzenberger 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.

AB - Let S be a transformation semigroup of degree n. To each element s is an element of 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 Schutzenberger 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.

KW - FINITE

UR - http://www.scopus.com/inward/record.url?scp=0032372698&partnerID=8YFLogxK

M3 - Article

SN - 0025-5874

VL - 228

SP - 435

EP - 450

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 3

ER -