Certain Equivalence Relations in Transformation Semigroups

The general object of this thesis is to study certain equivalence relations defined on a semigroup, in particular, to study certain equivalence relations defined on semigroups of single-valued transformations. We are interested in semigroups of transformations partly because every semigroup has as homomorphic image a semigroup of transformations (and hence a subsemigroup of a transformation semigroup of degree n for some n). This is a well-known fact, analogous to the Cayley Theorem on abstract groups, but we shall give a brief proof in Section 1. Section 1 is devoted to definitions and basic concepts. In Section 2 we prove some theorems concerning certain equivalence relations defined on a transformation semigroup of degree n. In Section 3 we present some results concerning the transformation semigroup T₃ of degree 3.

As an appendix, we have listed all subsemigroups of T₃ and the minimal generating sets of each such subsemigroup. The regular subsemigroups of T₃ are marked by an asterisk and the pseudo-inverses of each element of T₃ are listed.