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