Transformation Semigroups Over Groups

Abstract

The semigroup analogue of the symmetric group, S_{n}, is the full transformation semigroup, T_{n}. T_{n} is the set of all mappings from the set {1,2,..n} to itself. This semigroup has been studied in great detail, especially in connection with automata theory.

The wreath product of a group G by S_{n} has been studied for almost one hundred years. In this thesis, we study the wreath product of a group G by T_{n}. These wreath products are expressed as GwrS_{n} and GwrT_{n}, respectively. Many interesting theorems and properties for wreath products will be discussed. For example, the result of John Howie that every element in T_{n} − S_{n} can be expressed as a product of idempotents, is generalized to show that any element of GwrT_{n}- GwrS_{n} can be expressed as a product of idempotents.

It will also be shown that GwrT_{n} is unit regular.

Chapter five begins with a review of Green's relations for a moniod, M. Green's relations for T_{n} are also reviewed and R and L-classes for the wreath product GwrT_{n} are determined. Finally, in the last two chapters, the conjugacy class structures of GwrT_{n} are determined. Just as the conjugacy classes of GwrS_{n} are indexed by colored partitions, we show that the conjugacy classes of GwrT_{n} are indexed by certain colored directed graphs.