A study on d-system, m-system and n-system in ternary semigroups

Qualitative In this paper the terms d-system, m-system, n-system, U-ternary semigroup are introduced.  It is proved that an ideal A of a ternary semigroup T is a prime ideal of T if and only if T\A is an m-system of T or empty.  It is proved that an ideal A of a ternary semigroup T is completely semiprime if and only if T\A is a d-system of T or empty.  It is proved that every m-system in a ternary semigroup T is an n-system.  Further it is proved that an ideal Q of a ternary semigroup T is a semiprime ideal if and only if T\Q is an n-system of T (or) empty.  It is proved that if N is an n-system in a 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 a ternary semigroup T is U-ternary semigroup if and only if every ideal A of T is semiprime ideal of T.  Further it is proved that if T is a ternary semigroup and A is an ideal of T, then T is U-ternary semigroup if and only if T\A is an n-system of T or empty and if T is U-ternary semigroup and A is an ideal of T, then T\A is an m-system of T.