Certain Equivalence Relations in Transformation Semigroups

Author

Carol G. Doss, University of Tennessee - Knoxville

Date of Award

6-1955

Degree Type

Thesis

Degree Name

Master of Arts

Major

Mathematics

Major Professor

D. D. Miller

Committee Members

J. A. Cooley, Herbert L. Lee

Abstract

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.

Recommended Citation

Doss, Carol G., "Certain Equivalence Relations in Transformation Semigroups. " Master's Thesis, University of Tennessee, 1955.
https://trace.tennessee.edu/utk_gradthes/1120