Weakly Prime Ideals in Involution po-Γ-Semigroups

• Published in: Kyungpook mathematical journal Vol. 54; no. 4; pp. 629 - 638
• Main Authors: Abbasi, M.Y, Basar, Abul
• Format: Journal Article
• Language: Korean
• Published: 2014
• Subjects: po- ${\Gamma}$-semigroup; weakly prime ideal; involution
• ISSN: 1225-6951; 0454-8124

The concept of prime and weakly prime ideal in semigroups has been introduced by G. Szasz [4]. In this paper, we define the involution in po-${\Gamma}$-semigroups, then we extend some results on prime, semiprime and weakly prime ideals to the involution po-${\Gamma}$-semigroup S. Also, we characterize intra-regular involution po-${\Gamma}$-semigroups. We establish that in the involution po-${\Gamma}$-semigroup S such that the involution preserves the order, an ideal 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 ideals of S coincide. Finally, we prove that if S is a po-${\Gamma}$-semigroup with order preserving involution, then the ideals of S are prime if and only if S is intra-regular.