Weakly Prime Ideals in Involution po-Γ-Semigroups
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The concept of prime and weakly prime ideal in semigroups has been intro- duced by G. Szasz [4]. In this paper, we de ne the involution in po-Γ-semigroups, then we extend some results on prime, semiprime and weakly prime ideals to the involution po-Γ-semigroup S. Also, we characterize intra-regular involution po-Γ-semigroups. We establish that in the involution po-Γ-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-Γ-semigroup with order preserving involution, then the ideals of S are prime if and only if S is intra-regular.
Abstract
The concept of prime and weakly prime ideal in semigroups has been intro- duced by G. Szasz [4]. In this paper, we de ne the involution in po-Γ-semigroups, then we extend some results on prime, semiprime and weakly prime ideals to the involution po-Γ-semigroup S. Also, we characterize intra-regular involution po-Γ-semigroups. We establish that in the involution po-Γ-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-Γ-semigroup with order preserving involution, then the ideals of S are prime if and only if S is intra-regular.
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