A *simplicial poset* is a finite poset $P$ with minimial element $\hat{0}$ such that every interval $[\hat{0},x]$ is isomorphic to a Boolean lattice. Simplicial posets are generalizations of simplicial complexes (see, e.g., http://math.mit.edu/~rstan/pubs/pubfiles/82.pdf). Now, one way to define a matroid is as a simplicial complex for which the restriction to any subset of vertices is a pure complex. We could then naively define a simplicial poset $P$ to be *matroidal* if $P\setminus F$ is always graded, where $F$ is the order filter generated by any subset of atoms of $P$.

Have these matroidal simplicial posets been studied at all? Are there conjectures (e.g., about face numbers) for them?