Po-𝚪-Ideals in Po-𝚪-Semigroups

V. B. Subrahmanyeswara Rao Seetamraju · IOSR Journal of Mathematics · 2012

In this paper the terms; a completely prime po-Γ-ideal, c-system,a prime po-Γ-ideal, m-system of a po-𝚪-semigroup are introduced.It is proved that every po-𝚪-subsemigroup of a po-𝚪-semigroup is a c-system.It is also proved that a po-𝚪-ideal P of a po-𝚪-semigroup S is completely prime if and only if S\P is either a c-system or empty.It is proved that if P is a po-Γ-ideal of a po-Γ-semigroup S, then the conditions (1) if A, B are po-Γ-ideals of S and AΓB⊆P then either A⊆P or B⊆P, (2) if a, b ∈ S such that aΓS 1 Γb ⊆ P, then either a ∈ P or b ∈ P, are equivalent.It is proved that every completely prime po-𝚪-ideal of a po-𝚪-semigroup S is a prime po-𝚪-ideal of S. It is also proved that in a commutative po-𝚪-semigroup S, a po-𝚪-ideal P is a prime po-𝚪-ideal if and only if P is a completely prime po-𝚪-ideal.Further it is proved that a po-𝚪-ideal P of a po-𝚪-semigroup S is a prime po-𝚪-ideal of S if and only if S\P is an m-system or empty.In a globally idempotent po-Γ-semigroup, it is proved that every maximal po-Γ-ideal is a prime po-Γ-ideal.It is also proved that a globally idempotent po-Γ-semigroup having a maximal po-Γ-ideal, contains semisimple elements.The terms completely semiprime po-Γ-ideal,a semiprime po-Γ-ideal, n-system, d-system are introduced.It is proved that (1) every completely semiprime po-Γ-ideal of a po-Γ-semigroup is a semiprime po-Γ-ideal, (2) every completely prime po-Γ-ideal of a po-Γ-semigroup is a completely semiprime po-Γ-ideal.It is also proved that the nonempty intersection of any family of (1) completely prime po-Γ-ideals of a po-Γ-semigroup is a completely semiprime po-Γ-ideal, (2) prime po-Γ-ideals of a po-Γ-semigroup is a semiprime po-Γ-ideal.It is also proved that a po-Γ-ideal Q of a po-Γ-semigroup S is a semiprime iff S\Q is either an n-system or empty.Further it is proved that if N is an n-system in a po-Γ-semigroup S and a ∈ N, then there exists an m-system M of S such that a ∈ M and M ⊆ N. Mathematics Subject Clasification (2010) : 06F05, 06F99, 20M10, 20M99 Keywords: A po-Γ-semigroup, po-Γ-ideal, prime po-Γ-ideal, a completely prime po-Γ-ideal,a completely semiprime po-Γ-ideal, a semiprime po-Γ-ideal, po-c-system, po-d-system, po-m-system, po-n-system. I .

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