Let $\mathcal{X}$ be a family of subsets of a finite set E. A matroid on E is called an $\mathcal{X}$-matroid if each set in $\mathcal{X}$ is a circuit. We develop techniques for determining when there exists a unique maximal $\mathcal{X}$-matroid in the weak order poset of all $\mathcal{X}$-matroids on E and formulate a conjecture which would characterise the rank function of this unique maximal matroid when it exists. The conjecture suggests a new type of matroid rank function which extends the concept of weakly saturated sequences from extremal graph theory. We verify the conjecture for various families $\mathcal{X}$ and show that, if true, the conjecture could have important applications in such areas as combinatorial rigidity and low rank matrix completion.

让$\mathcal{X}$是有限集E的子集族。E上的拟阵称为$\mathcal{X}$-maroid 如果每个设置$\mathcal{X}$是一个电路。我们开发技术来确定何时存在唯一的最大值$\mathcal{X}$-所有弱阶偏序集中的拟阵$\mathcal{X}$-E上的拟阵并制定一个猜想，该猜想将描述该唯一最大拟阵存在时的秩函数。该猜想提出了一种新型拟阵秩函数，它扩展了极值图论中弱饱和序列的概念。我们为各家验证猜想$\mathcal{X}$并表明，如果正确，该猜想可能在组合刚性和低秩矩阵完成等领域具有重要的应用。