A STUDY ON STRUCTURE OF PO-TERNARY SEMIRINGS

Abstract

This paper is divided into two sections. In section 1, the notion of a PO-ternary semiring was introduced and examples are given.  Further the terms commutative PO-ternary semiring , quasi commutative PO-ternary semiring, normal PO-ternary semiring, left pseudo commutative PO-ternary semiring, lateral  pseudo commutative PO-ternary semiring, right pseudo commutative PO-ternary semiring and pseudo commutative PO-ternary semiring are introduced and characterized them.  Further the terms left singular, right singular and singular with respect to addition and left singular, right singular, lateral singular, singular with respect to ternary multiplication and two sided singular are introduced and made a study on them. In section 2, the terms; PO-ternary subsemiring, PO-ternary subsemiring of T generated by a set A, cyclic PO-ternary subsemiring and cyclic PO-ternary semiring are introduced.  It is proved that T be a PO-ternary semiring and A be a non-empty subset of T.  Then (A) = {a₁a₂....a_n-1;n ϵ N, a₁,a₂....a_n Ïµ A } is a smallest PO-ternary subsemiring of T.  Let T be a PO-ternary semiring and A be a non-empty subset of T.  <A> = the intersection of all PO-ternary subsemirings of  T containing A.