A Note on One Sided and Two Sided PO-Ternary Ideals in PO-Ternary Semiring

Dasari Madhusudhana Rao; Pusapati Siva Prasad; G Srinivasa Rao

Dasari Madhusudhana Rao Head, Department of Mathematics, VSR & NVR College, Tenali, Guntur (Dt), Andhra Pradesh, India.
Pusapati Siva Prasad Asst. Prof of Mathematics, Universal College of Engineering & Technology Perecherla, Guntur, A. P. India
G Srinivasa Rao Asst. Prof of Mathematics, Tirumala Engineering College, Narasaraopet, A. P. India

Keywords: PO-ternary Semiring, left PO-ternary ideal, lateral PO-ternary ideal, right PO-ternary ideal, two sided PO-ternary ideal, left simple, lateral simple, right simple.

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) = (T^e T^ea + na] = (T^e T^ea U na] (M(a) = (T^ea T^e + T^eT^e aT^eT^e + na] = ( T^ea T^e U T^eT^e aT^eT^e U na], R (a) = (aT^e T^e + na ] = (aT^e T^e U na] and T(a) = (T^e T^ea + aT^e T^e + T^eT^e aT^eT^e +na] = (T^e T^ea UaT^e T^eUT^eT^e aT^eT^e 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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