On the Rank Functions of Powerful Sets

TL;DR

This paper explores the structural properties of powerful sets, a generalization of binary matroids, by analyzing their rank functions and characterizing degenerate elements, confirming two conjectures by Farr and Wang.

Contribution

It characterizes powerful sets through their rank functions, introduces powerful multisets, and proves that a powerful set is a binary matroid if and only if its rank function is subcardinal.

Findings

Rank function evaluations determine degenerate elements.

Powerful sets correspond to binary matroids iff their rank function is subcardinal.

Confirmed Farr and Wang's two conjectures.

Abstract

A set $S\subseteq 2^E$ of subsets of a finite set $E$ is \emph{powerful} if, for all $X\subseteq E$, the number of subsets of $X$ in $S$ is a power of 2. Each powerful set is associated with a non-negative integer valued function, which we call the rank function. Powerful sets were introduced by Farr and Wang as a generalisation of binary matroids, as the cocircuit space of a binary matroid gives a powerful set with the corresponding matroid rank function. In this paper we investigate how structural properties of a powerful set can be characterised in terms of its rank function. Powerful sets have four types of degenerate elements, including loops and coloops. We show that certain evaluations of the rank function of a powerful set determine the degenerate elements. We introduce powerful multisets and prove some fundamental results on them. We show that a powerful set corresponds to a binary matroid if and only if its rank function is subcardinal. This paper answers the two conjectures made by Farr and Wang in the affirmative.