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

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