Theory of PO-Ternary Ideals in PO-Ternary Semiring

In this paper, the terms PO-ternary ideal, proper PO-ternary ideal, trivial PO-ternary ideal, maximal PO-ternary ideal, globally idempotent PO-ternary ideal, globally idempotent PO-ternary semiring, PO-ternary ideal of T generated by a set A, principal PO-ternary ideal, simple PO-ternary semiring, semi-simple, semisimple PO-ternary semiring are introduced. It is proved that (1) The nonempty intersection of any family of PO-ternary ideals of a PO-ternary semiring T is a PO-ternary ideal of T. (2) The union of any family of PO-ternary ideals of a PO-ternary semiring T is a PO-ternary ideal of T. (3) If A is a PO-ternary ideal of a PO-ternary semiring T with multiplicative identity e and e A then A = T. (4) If T is a PO-ternary semiring with multiplicative identity e then the union of all proper ideals of T is the unique maximal ideal of T. (5) The PO-ternary ideal of a PO-ternary semiring T generated by a non-empty subset A is the intersection of all PO-ternary ideals of T containing A. (6) If T is a PO-ternary semiring and aT then J(a) = (A], where A = and denotes a finite sum and is the set of all positive integer with zero. (7) Let A, B be two PO-ternary ideals of PO-ternary semiring T with zero and let A + B = { a + b : a∈ A, b∈ B}. Then A + B is a PO-ternary ideal generated by (A ∪ B). (8) The left PO-ternary ideal generated by the union A ∪ B of two left PO-ternary ideals is the set A + B consisting of the elements of PO-ternary semiring T obtained on adding any element of A to any element of B. (9) In any PO-ternary semiring T, the following are equivalent. Principal PO-ternary ideals of T form a chain. PO-ternary Ideals of T form a chain. (10) If T is a left simple PO-ternary semiring or a lateral simple PO-ternary semiring or a right simple PO-ternary semiring then T is a simple PO-ternary semiring. (11) A PO-ternary semiring T is simple PO-ternary semiring if and only if (TTaTT] = T for all aT. (12) A PO-ternary semiring T is regular then every principal PO-ternary ideal of T is generated by an idempotent. (13) An element a of a PO-ternary semiring T is said to be semisimple if a(] i.e. (] = (< a >] for all odd natural number n. (14) Let T be a PO-ternary semiring and . If a is regular then a is semisimple. (15) a be an element of a PO-ternary semiring T. If a is left regular or lateral regular or right regular, then a is semisimple. (16) Let a be an element of a PO-ternary semiring T. If a is intra regular then a is semisimple.