References.

• S. A. Linton, G. Pfeiffer, E. F. Robertson and N. Ruskuc, Groups and Actions in Transformation Semigroups. Math. Z. 228 (1998), 435-450.

  Abstract.

  Let S be a transformation semigroup of degree n. To each element s in 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.

  Available as DVI file (67 kB) and as compressed PostScript file (117 kB).

• S. A. Linton, G. Pfeiffer, E. F. Robertson and N. Ruskuc, Computing Transformation Semigroups. J. Symbolic Comput. 33 (2002), 145--162.

  Abstract.

  This paper describes algorithms for computing the structure of finite transformation semigroups. The algorithms depend crucially on a new data structure for an R-class in terms of a group and an action. They provide for local computations, concerning a single R-class, without computing the whole semigroup, as well as for computing the global structure of the semigroup. The algorithms have been implemented in the share package MONOID within the GAP system for computational algebra.
  Available as DVI file (75 kB) and as compressed PostScript file (94 kB).

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