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

TL;DR

This paper characterizes the smallest ideal and its closure in the semigroup ($eta$N,.) by showing they form single equivalence classes under extended divisibility relations, revealing their algebraic structure.

Contribution

It proves that all elements in the smallest ideal and its closure in ($eta$N,.) form single equivalence classes under extended divisibility relations, clarifying their algebraic properties.

Findings

All elements in the smallest ideal form a single equivalence class.

All elements in the closure of the smallest ideal form a single equivalence class.

The results deepen understanding of the algebraic structure of ($eta$N,.)

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.