Normalized Matching in Direct Products of Partial Orders

Abstract

In this paper we investigate the question: When does the direct product of partial orders each satisfying the normalized matching condition also satisfy it? A proof of Harper's result that a sufficient condition for this is that factor partial orders have rank populations that are multiplicatively convex, is presented. A general necessary and sufficient condition is described, and conditions which occur when one factor is a chain are obtained. In particular we show that under these circumstances it is necessary that the product order have multiplicatively convex population rank.