#### 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 =∼.

#### OCLC Number

1090140805

#### Recommended Citation

Khalifa, Salahddeen, "Divisibility in the Stone-Cech Compactiﬁcation of N" (2018). *Dissertations*. 792.

https://irl.umsl.edu/dissertation/792