How is a directed simplicial complex definable as a collection of ordered sets? Reading this paper. It says that
"An abstract directed simplicial complex is a collection $S$ of finite, ordered sets with the property that if $\sigma \in S$, then every subset $\tau$ of $\sigma$, with the natural ordering inherited from $\sigma$, is also a member of $S$."
If I understand correctly, the elements of the ordered sets are the vertices of the simplices. But how could an ordering encapsulate all possible ways of directing the edges?
Take a triangle for example, $ABC$. To say $A<B<C$ could be thought of as directing the edge AB in the direction of A, BC in the direction of B, etc. But how could an ordering describe a directed triangle where one vertex has no edges coming into it? Or no edges going out of it?
 A: It's easy for an ordering to "describe a directed triangle where one vertex has no edges coming into it, or no edges going out of it": if we order $A < B < C$ and have each edge go in the direction of the smaller endpoint, then $A$ has no edges going out of it, and $C$ has no edges coming into it. The thing you can't do with an ordering is create cycles: cases where the edges are oriented $A \to B \to C \to A$ or $A \to C \to B \to A$.
The idea (in this paper; I don't know if this usage of directed simplices is conventional) is that a directed simplex is by definition acyclic, in which case an ordering suffices. 
You can represent any directed graph as a directed simplicial complex by taking each edge to be a $2$-simplex directed in the appropriate direction. So if you wanted a cycle on vertices $A,B,C$, you could have the $2$-simplex $\{A,B\}$ with $A<B$, the $2$-simplex $\{B,C\}$ with $B<C$, and the $2$-simplex $\{C,A\}$ with $C<A$. The inheritance property merely says that $\{A\}$, $\{B\}$, and $\{C\}$ have to be $1$-simplices, and there's nothing nontrivial to say about the ordering there.
In this directed simplicial complex, you couldn't add the $3$-simplex $\{A,B,C\}$, because there's no consistent ordering to put on it. Conversely, if your directed simplicial complex contains $\{A,B,C\}$ with $A<B<C$, then it must contain the $2$-simplices $\{A,B\}$ with $A<B$, $\{A,C\}$ with $A<C$, and $\{B,C\}$ with $B<C$, as well as the $1$-simplices $\{A\}$, $\{B\}$, and $\{C\}$.
What I think the authors are doing here when they say things like

the reconstructions consistently contained directed simplices of dimensions up to $6$ or $7$

is taking a directed graph and constructing the simplicial complex in which every acyclic directed clique is a simplex. This clearly satisfies the definition of a directed simplicial complex: if a set of vertices $\sigma$ induces an acyclic directed clique, and $\tau \subset \sigma$, then $\tau$ also induces an acyclic directed clique.
