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An Introduction of Theory of Involutions in Ordered Semihypergroups and their Weakly Prime Hyperideals


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1 Department of Mathematics, Jamia Millia Islamia, New Delhi - 110 025, India
     

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In this paper, we introduce ordered semihypergroups with involution and weakly prime semihyperideal, then we investigate some properties of prime, semiprime and weakly prime hyperideals in ordered semihypergroup with involution. Also, we study intra-regular ordered semihypergroups with involution. We prove that in ordered semihypergroup S with involution such that the involution preserves the order, a semihyperideal of S is prime if and only if it is both weakly prime and semiprime and if S is commutative, then the prime and weakly prime hyperideals of S coincide. Finally, we show that if S is an ordered semi- hypergroup with order preserving involution, then the semihyperideals of S are prime if and only if it is intra-regular.

Keywords

Semigroup, Ordered Semihypergroup, Hyperideal, Involution, Weakly Prime Hyperideal.
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  • An Introduction of Theory of Involutions in Ordered Semihypergroups and their Weakly Prime Hyperideals

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Authors

Abul Basar
Department of Mathematics, Jamia Millia Islamia, New Delhi - 110 025, India
Mohammad Yahya Abbasi
Department of Mathematics, Jamia Millia Islamia, New Delhi - 110 025, India
Sabahat Ali Khan
Department of Mathematics, Jamia Millia Islamia, New Delhi - 110 025, India

Abstract


In this paper, we introduce ordered semihypergroups with involution and weakly prime semihyperideal, then we investigate some properties of prime, semiprime and weakly prime hyperideals in ordered semihypergroup with involution. Also, we study intra-regular ordered semihypergroups with involution. We prove that in ordered semihypergroup S with involution such that the involution preserves the order, a semihyperideal of S is prime if and only if it is both weakly prime and semiprime and if S is commutative, then the prime and weakly prime hyperideals of S coincide. Finally, we show that if S is an ordered semi- hypergroup with order preserving involution, then the semihyperideals of S are prime if and only if it is intra-regular.

Keywords


Semigroup, Ordered Semihypergroup, Hyperideal, Involution, Weakly Prime Hyperideal.

References