• PARTIALLY ORDERED FILTERS IN PARTIALLY ORDERED SEMIGROUPS

Abstract

In this paper the terms, po filter, po filter of a po semigroup generated by a subset and principle po filter generated by an element a  in a po semigroup  are introduced.   It is proved that in a po semigroup S, the nonempty intersection of a family of po filters is also a po filter of S.  It is also proved that a nonempty subset F of a po semigroup S is a po filter if and only if  S\F is a completely prime po ideal of S or empty. It is proved that every po filter F of a po semigroup S is a i) po-c-system of S  ii) po-m-system of S  and iii) po-d-system of S.  It is proved that the po filter of a po semigroup S generated by a nonempty subset A is the intersection of all po filters of S containing A.  Let S be a po semigroup and a ∈ S, then it is proved that N(a)  is the least filter of S containing {a}.