Representing the Smallest Ideal and it's closure in the semigroup ($\beta$N,.) as equivalence classes under the extension of divisibility relations

Abstract

In this paper we will prove that all the elements in the smallest ideal K($\beta$N) in the semigroup of the Stone Cech compactification ($\beta$N,.) of the discrete semigroup of natural numbers N under multiplication constitute a single equivalence class under the mid extension divisibility relation. And all the elements in the closure of the smallest ideal Cl($\beta$N) constitute a single equivalence class under the tilde extension divisibility relation.