Maximal Matroids in Weak Order Posets

TL;DR

This paper investigates conditions for the existence and uniqueness of maximal matroids constrained by a family of circuits within the weak order poset, and characterizes their rank functions.

Contribution

It provides criteria for the existence and uniqueness of maximal X-matroids and describes their rank functions when they exist.

Findings

Criteria for the existence of a unique maximal X-matroid.

Characterization of the rank function of the maximal X-matroid.

Insights into the structure of X-matroids in the weak order poset.

Abstract

Let $\cX$ be a family of subsets of a finite set $E$. A matroid on $E$ is called an $\cX$-matroid if each set in $\cX$ is a circuit. We consider the problem of determining when there exists a unique maximal $\cX$-matroid in the weak order poset of all $\cX$-matroids on $E$, and characterizing its rank function when it exists.