Proximity Induced by Order Relations

This paper introduces an order proximity on a collection of objects induced by a partial order using the Smirnov closeness measure on a Sz\'{a}z relator space. A Sz\'{a}z relator is a nonempty family of relations defined on a nonvoid set $K$. The Smirnov closeness measure provides a straightforward means of assembling partial ordered of pairwise close sets. In its original form, Ju. M. Smirnov closeness measure $\delta(A,B) = 0$ for a pair of nonempty sets $A,B$ with nonvoid intersection and $\delta(A,B) = 1$ for non-close sets. A main result in this paper is that the graph obtained by the proximity is equivalent to the Hasse diagram of the order relation that induces it. This paper also includes an application of order proximity in detecting sequences of video frames that have order proximity.