Antimatroid that is not a partial order

Question

I am trying to understand the difference between an antimatroid on a set and a partial order on a set. According to wikipedia, an antimatroid is a generalization of partial order. Thus, for some set $S$, there must be some antimatroid on $S$ for which there is no corresponding partial order in $S$. Can someone give me an example of this? What is the smallest $|S|$ where this can occur?

Answer 1

In any matroid that comes from a partial order, the feasible sets are closed under intersections, since the lower sets in a poset are closed under intersections. As an example where this fails, consider the antimatroid on the set $S=\{a,b,c\}$ in which every subset except $\{c\}$ is feasible. This antimatroid cannot come from a partial order since $\{a,c\}$ and $\{b,c\}$ are feasible but their intersection is not.