# Matroidal simplicial posets?

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?

• Are they Cohen-Macaulay posets? Is the ring $A_P$ or $\tilde{A}_P$ of your reference a level algebra? Apr 5, 2019 at 15:58
• @RichardStanley: It might make sense to restrict to the Cohen-Macaulay case. I was in particular thinking there might be an analog of your matroid $h$-vector conjecture here. Apr 5, 2019 at 16:05
• (It might even make sense to restrict to Gorenstein* $P$.) Apr 5, 2019 at 16:29
• Are there "interesting" Gorenstein* matroidal simplicial posets? The only Gorenstein* geometric lattices are boolean algebras. Apr 5, 2019 at 21:37