Document Type

Dissertation

Degree

Doctor of Philosophy

Major

Applied Mathematics

Date of Defense

11-8-2018

Graduate Advisor

Dr. Ronald Dotzel

Committee

Dr. Prabhakar Rao

Dr. Ravindra Girivaru

Dr. Adrian Clingher

Abstract

Let S a discrete semigroup. The associative operation on S extends naturally to an associative operation on βS,the Stone Cech compactification of S. This involves both topology and algebra and leads us to think how to extend properties and operations that are defined on S to βS. A good application of this is the extension of relations and divisibility operations that are defined on the discrete semigroup of natural numbers (N,.) with multiplication as operation to relations and divisibility operations that are defined on (βN,?) where (?) is the extension of the operation (.). In this research I studied extending the usual divisibility relation | that is defined on N with multiplication to the divisibility relations : |l,|r,|m and ˜ | which are definedon βN. I divided the elements of βN into ultrafilters which are on finite levels and ultrafilters which are not on finite levels. That helps me to work more accurately with elements of βN to get good results about the extension of divisibility relations. Moreover I represented all elements in the smallest ideal K(βN) in the semigroup (βN,?) by a single equivalence class under the relation =m and all elements in the closure of the smallest ideal CL(K(βN)) by a single equivalence class under the relation =∼.

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