On certain Γ-hyperideals in Γ-semihypergroups

In this paper pseudo symmetric Γ-hyperideal in Γsemihypergroups is introduced and characterized. It is proved thatthe class of pseudo symmetric Γ-semihypergroups contains left (right) duo Γ-semihypergroups, quasi commutative Γ-semihypergroups with unity, left (right) pseudo commutative Γ-semihypergroups and idempotent Γ-semihypergroups. The notions of completely prime Γhyperideal, semiprime Γ-hyperideal, partially semiprime Γ- hyperideal are also defined and completely prime Γ-hyperideal of a Γsemihypergroup has been characterized in terms of prime Γ- hyperideal and pseudo symmetric Γ-hyperideal. The characterization of completely semiprime Γ-hyperideals of Γ-semihypergroupsis presented. In Γ-semihypergroup n-semiprime Γ-hyperideal and npartially semiprime Γ-hyperideal are introduced as a generalization of semiprime Γ-hyperideal and partially semiprime Γ-hyperideal of Γ-semihypergroup respectively. The notion of semi-extension of Γhyperideal in Γ-semihypergroup has also been defined. Some related results are proved along with establishing the relationship between n-semiprime Γ-hyperideals and semi-extension of Γ-hyperideal in commutative Γ-semihypergroups.

Keywords:

Γ-semihypergroup , Γ-hyperideal , Pseudo Symmetric Γ-hyperideal , Completely Prime Γ-hyperideal , n-semiprime Γ-htperideal