Let $E$ be a finite set, and let $\mathcal F \subseteq \mathcal P(E)$ be a collection of subsets of $E$ satisfying the following three axioms:

1. $E \in \mathcal F$. 2. If $\mathcal G \subseteq \mathcal F$ is nonempty, then $\bigcap_{F \in \mathcal G} F \in \mathcal F$. 3. For every $F \in \mathcal F$, let \begin{align*} \mathcal C(F) := \{G \in \mathcal F : F \subsetneq G \text{ and there is no } H \in \mathcal F \text{ with } F \subsetneq H \subsetneq G\}. \end{align*}

Then the family $\{G \setminus F : G \in \mathcal C(F)\}$ is a partition of $E \setminus F$.

Then there exists a matroid $M$ on ground set $E$ whose collection of flats is exactly $\mathcal F$.