Convex and Weakly Convex Subsets of a Pseudo Ordered Set

Abstract

In this paper the notion of convex and weakly convex (w-convex) subsets of a pseudo ordered set is introduced and several characterizations are proved. It is proved that set of all convex subsets of a pseudo ordered set $A$ forms a complete lattice. Notion of isomorphism of psosets is introduced and characterization for convex isomorphic psosets is obtained. It is proved that lattice of all w-convex subsets of a pseudo ordered set $A$ denoted by $WCS(A)$ is lower semi modular. Also we have proved that for any two pseudo ordered sets $A$ and $A^1$, w-convex homomorphism maps atoms of $WCS(A)$ to atoms of $WCS(A^1)$. Concept of path preserving mapping is introduced in a pseudo ordered set and it is proved that every mapping of a pseudo ordered set $A$ to itself is path preserving if and only if $A$ is a cycle.