Let $\mathcal{F}$ be a collection of subsets of a finite set $X$, partially ordered by inclusion, and let $\mathcal{I}$ consist of the antichains of $\mathcal{F}$, taken as independent sets.
Under what conditions on $\mathcal{F}$ does $\mathcal{I}$ form a matroid structure on $\mathcal{F}$?