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Let $E$ be a set, and $F \in \mathcal P(E)$ has the following property:

For every $x\in E$ and $Y,Z\in F$ with $x\notin Y\cup Z$, there exists $X\in F$ with $(Y\cap Z)\cup\{x\}\subseteq X$.

I wonder if this property has a name? What kind of set system is defined to have the property?

An example is from matroid theory:

The family $\mathcal{H}$ of hyperplanes of a matroid has the following properties, which may be taken as yet another axiomatization of matroids:[4]

  • There do not exist distinct sets $X$ and $Y \in \mathcal{H}$ with $X\subset Y$. That is, the hyperplanes form a Sperner family.

  • For every $x\in E$ and $Y,Z\in\mathcal{H}$ with $x\notin Y\cup Z$, there exists $X\in\mathcal{H}$ with $(Y\cap Z)\cup\{x\}\subseteq X$.

Thanks.

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I believe it is known as one of the two hyperplane axioms. For reference, please see Matroid Theory by D.J.A. Welsh, pages 39-40. Here is the link to it on google books (and pages 39-40 are freely available in the preview). The book then seems to go on to cover some other systems with this property, but the next few pages are not freely available.

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