A Note on One Sided and Two Sided PO-Ternary Ideals in PO-Ternary Semiring
Abstract
In this paper the term, left(lateral, right and two sided) PO-ternary ideal, maximal left(lateral, right and two sided) PO-ternary ideal, left (lateral, right and two sided) PO-ternary ideal of T generated by a set A, principal left (lateral, right and two sided) PO-ternary ideal generated by an element a left (lateral, right and two sided) simple PO-ternary semiring are introduced. It is proved that (1) the non-empty intersection of any two left (lateral, right and two sided) PO-ternary ideals of a PO-ternary semiring T is a left (lateral, right and two sided) PO-ternary ideal of T. (2) non-empty intersection of any family of left (lateral, right and two sided) PO-ternary ideals of a POternary semiring T is a left(lateral, right and two sided) PO-ternary ideal of T. (3) the union of any left PO-ternary ideals of a PO-ternary semiring T is a left PO-ternary ideal of T. (4) the union of any family of left(lateral, right and two sided) PO-ternary ideals of a PO-ternary semiring T is a left(lateral, right and two sided) PO-ternary ideal of T. (5) The left (lateral, right and two sided) PO-ternary ideal of a PO-ternary semiring T generated by a non-empty subset A is the intersection of all left(lateral, right and two sided) PO-ternary ideals of T containing A. (6) If T is a PO-ternary semiring and a∈T then L(a) = (Te Tea + na] = (Te Tea U na] (M(a) = (Tea Te + TeTe aTeTe + na] = ( Tea Te U TeTe aTeTe U na], R (a) = (aTe Te + na ] = (aTe Te U na] and T(a) = (Te Tea + aTe Te + TeTe aTeTe +na] = (Te Tea UaTe Te UTeTe aTeTe Una]). (7) A PO-ternary semiring T is a left(lateral, right) simple PO-ternary semiring if and only if (TTa] = T ((TaT U TTaTT] = T, (aTT] = T) for all a∈T.
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