Date of Award

8-1949

Degree Type

Thesis

Degree Name

Master of Arts

Major

Mathematics

Major Professor

D. D. Wilson

Committee Members

J. A. Cooley, Alan H. Neilsen

Abstract

The purpose of this thesis is to exhibit the direct products and the semidirect products of semigroups of orders two, three, and four. Section I consists of the definitions necessary for the proofs and computations which appear in the first five sections. In Section 2 theorems on direct products of semigroups of any order are proved. A method of writing by inspection the multiplication table of the direct product of two finite semigroups is presented in Section 3. Section 4 consists of tables listing the direct products of semigroups of order two with semigroups of orders two, three and four and a table listing the direct products of semigroups of order three with semigroups of order three. In Section 5 is considered the problem of describing the multiplication table of the direct product of two semigroups S and T by adjoining rows and columns to the multiplication table of the direct products of subsemigroups contained in S and T respectively. The direct products of those third and fourth order semigroups which are not so described are listed in Section 5. In Section 6, semidirect products of semigroups are defined, some theorems concerning them are proved, and examples are cited to show that some of the theorems on direct products are not necessarily true for semidirect products. A method of writing by inspection the multiplication table of the semidirect product of a finite semigroup and the semigroup of its endomorphisms is exhibited in Section 7. The endomorphisms of the semigroups of orders two and three, the automorphisms of the semigroups of order four, and the semidirect products of orders two, three, and four are listed in Section 8.

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Mathematics Commons

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