Abstract

In this paper, the terms, ‘A-potent’, ‘left A-divisor’, ‘right A-divisor’, ‘A-divisor’ elements, ‘N(A)-ternary semigroup’ for an ideal A of a ternary semigroup are introduced. If A is an ideal of a ternary semigroup T then it is proved that (1)  (2) N₀(A) = A₂, N₁(A) is a semiprime ideal of T containing A, N₂(A) = A₄ are equivalent, where N_o(A) = The set of all A-potent elements in T, N₁(A) = The largest ideal contained in N_o(A), N₂(A) = The union of all A-potent ideals. If A is a semipseudo symmetric ideal of a ternary semigroup then it is proved that N₀(A) = N₁(A) = N₂(A). It is also proved that if A is an ideal of a ternary semigroup such that N₀(A) = A then A is a completely semiprime ideal. Further it is proved that if A is an ideal of ternary semigroup T then R(A), the divisor radical of A, is the union of all A-divisor ideals in T.  In a N(A)- ternary semigroup it is proved that R(A) = N₁(A). If A is a semipseudo symmetric ideal of a ternary semigroup T then it is proved that S is an N(A)- ternary semigroup iff R(A) = N₀(A).  It is also proved that if M is a maximal ideal of a ternary semigroup T containing a pseudo symmetric ideal A then M contains all A-potent elements in T or T\M is singleton which is A-potent.

Key words:  Pseudo symmetric ideal, semipseudo symmetric ideal, prime ideal, semiprime ideal, completely prime ideal, completely semiprime ideal, semisimple element, A-potent element, A-potent Γ-ideal, A-divisor, N(A)- ternary semigroup.