There are many ways to define directed hypergraphs. More precisely, there are many ways to define directed hyperedges (or hyperarcs). A quite general definition taken from  is the following.
Given a finite node set V, a directed hyperedge (or hyperarc) above V is an ordered pair (T,H) of disjoint subsets of V. The set T is the tail set of the hyperarc, while H is called the head set of the hyperarc. A directed hypergraph is a pair (V,A), where V is a finite node set and A is a set of hyperarcs above V.
When studying connectivity properties of directed hypergraphs, we can assume that neither the tail set, nor the head set is empty in a hyperarc. Sometimes the notion "directed hypergraph" is used for the restricted case where each hyperarc has only one head (called B-graphs in ).