• COMPLETELY PRIME PO IDEALS AND PRIME PO IDEALS IN PO TERNARY SEMIGROUPS

V. Siva Rami Reddy, V. Sambasiva Rao, A. Anjaneyulu, A. Gangadhara Rao*

Abstract


In this paper the terms, completely prime ideal, prime ideal, completely semiprime ideal, semiprime ideal, prime radical and complete prime radical in a po ternary semigroup are introduced. It is proved that in a po ternary semigroup (i) A is a prime ideal of T, (ii) For a, b, c ∈ T; < a > < b > < c > ⊆ A implies a ∈ A or b ∈ A or c ∈ A,
(iii) For a; b; c ∈ T; T1T1aT1T1b T1T1c T1T1 ⊆ A  implies a ∈ A or b ∈ A or c ∈ A are equivalent.  It is proved that A A po ternary ideal P of a po ternary semigroup T is (1) completely prime iff T\P is either a po ternary subsemigroup of T or empty (2) prime iff T\P is either an m-system or empty.  It is also proved that every completely prime ideal of a po ternary semigroup is prime.  In a globally idempotent po ternary semigroup, it is proved that every maximal ideal is prime.  It is also proved that a globally idempotent po ternary semigroup having a maximal ideal contains semisimple elements.  It is proved that a po ternary ideal A of a po ternary semigroup T is completely semiprime if and only if
x ∈ T, x3 ∈ A implies x ∈ A. It is proved that if A is a completely semiprime ideal of a po ternary semigroup T, then
x, y, z ∈ T, xyz ∈ A implies that xyTTz ⊆ A,  xTTyz ⊆ A and xTyTz ⊆ A.  It is also proved that every completely semiprime ideal of a po ternary semigroup is semiprime. It is proved that a po ternary ideal A of a po ternary semigroup T is completely semiprime if and only if T\A is a d-system of T or empty.  It is also proved that the nonempty intersection of a family of (1) completely prime ideals of a po ternary semigroup is completely semiprime (2) prime ideals of a po ternary semigroup is semiprime. And also proved that a po ternary ideal Q of a semigroup T is
(1) semiprime iff T\Q is either an n-system or empty.  It is proved that if N is an n-system in a po ternary semigroup T and a N, then there exist an m-system M in T such that a M and M N.  It is proved that to each ideal A of a semigroup T,

Keywords


completely prime ideal, prime ideal, completely semiprime ideal, semiprime ideal, prime radical and complete prime radical.

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