A Note On Archimedian Chained Γ-Semigroups


Authors : Poonam Kumar Sharma, Bhavanari Satyanarayana, Abul Basar

Volume/Issue : Volume 4 - 2019, Issue 12 - December

Google Scholar : https://goo.gl/DF9R4u

Scribd : https://bit.ly/34XwGji

Abstract : In this article, we introduce chained Γ-semigroups, cancellative Γ-semigroups and obtain some equivalent conditions. Also, we prove that if S is a chained Γ -semigroup, then S is an Archimedian Γ-semigroup with no Γ-idempotents if and only if s ωΓS satisfies the concentric condition for every s ∈ S . Furthermore, we prove that a cancellative Archimedian chained Γ -semigroup is a Γ -group if s ωΓS does not satisfy the concentric condition for some s ∈ S. Finally, we prove that if S is a chained Γ-semigroup containing cancellable elements. Then, S is a cancellative Γ -semigroup provided s ωΓS satisfies the concentric condition for every s ∈ S. The converse is true if S is a Noetherian Γ-semigroup without Γ-idempotents.

Keywords : Maximal Γ -Ideal, Prime Γ -Ideal, Γ -Radical, Γ -Idempotent, Chained Γ -Semigroup , Archimedian Γ-Semigroup, Noetherian Chained Γ-Semigroup.

In this article, we introduce chained Γ-semigroups, cancellative Γ-semigroups and obtain some equivalent conditions. Also, we prove that if S is a chained Γ -semigroup, then S is an Archimedian Γ-semigroup with no Γ-idempotents if and only if s ωΓS satisfies the concentric condition for every s ∈ S . Furthermore, we prove that a cancellative Archimedian chained Γ -semigroup is a Γ -group if s ωΓS does not satisfy the concentric condition for some s ∈ S. Finally, we prove that if S is a chained Γ-semigroup containing cancellable elements. Then, S is a cancellative Γ -semigroup provided s ωΓS satisfies the concentric condition for every s ∈ S. The converse is true if S is a Noetherian Γ-semigroup without Γ-idempotents.

Keywords : Maximal Γ -Ideal, Prime Γ -Ideal, Γ -Radical, Γ -Idempotent, Chained Γ -Semigroup , Archimedian Γ-Semigroup, Noetherian Chained Γ-Semigroup.

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