Conjugacy and Iteration of Standard Interval Rank in Finite Ordered Sets

TL;DR

This paper explores the properties and convergence of interval rank functions in finite posets, introducing conjugate orders and iterative strategies to extend rank concepts beyond graded structures.

Contribution

It introduces conjugate interval orders and analyzes the iterative application of interval rank functions, advancing the understanding of rank extension in arbitrary finite posets.

Findings

Conjugate orders partition pairwise comparisons of interval elements.

Iterative application of interval ranks shows convergence properties.

Experimental mathematics supports theoretical insights.

Abstract

In order theory, a rank function measures the vertical "level" of a poset element. It is an integer-valued function on a poset which increments with the covering relation, and is only available on a graded poset. Defining a vertical measure to an arbitrary finite poset can be accomplished by extending a rank function to be interval-valued. This establishes an order homomorphism from a base poset to a poset over real intervals, and a standard (canonical) specific interval rank function is available as an extreme case. Various ordering relations are available over intervals, and we begin in this paper by considering conjugate orders which "partition" the space of pairwise comparisons of order elements. For us, these elements are real intervals, and we consider the weak and subset interval orders as (near) conjugates. It is also natural to ask about interval rank functions applied reflexively on whatever poset of intervals we have chosen, and thereby a general iterative strategy for interval ranks. We explore the convergence properties of standard and conjugate interval ranks, and conclude with a discussion of the experimental mathematics needed to support this work.