Operations on Partial Orders

TL;DR

This paper introduces Boolean-like operations on partial orders, including set-based operations and height-based probability measures, to extend the algebraic and probabilistic analysis of such structures.

Contribution

It develops analogues of Boolean operations for partial orders and introduces height-based probability measures, expanding the tools for analyzing incomplete ordered structures.

Findings

Set operations on partial orders often yield sets of elements.

Adding elements to sets based on sup and inf is useful even when these do not exist.

Height of elements can be used to define probability measures.

Abstract

We define analogues of Boolean operations on not necessarily complete partial orders, they often have as results sets of elements rather than single elements. It proves useful to add to such sets X if they are intended to be sup(X) or inf(X), even if sup and inf do not always exist. We then define the height of an element as the maximal length of chains going from BOTTOM to that element, and use height to define probability measures.