  Transit functions were introduced as models of betweenness on undirected
structures. Here we introduce directed transit function as the directed
analogue on directed structures such as posets and directed graphs. We first
show that betweenness in posets can be expressed by means of a simple set of
first order axioms. Similar characterizations can be obtained for graphs with
natural partial orders, in particular, forests, trees, and mangroves. Relaxing
the acyclicity conditions leads to a generalization of the well-known geometric
transit function to the directed structures. Moreover, we discuss some
properties of the directed analogues of prominent transit functions, including
the all-paths, induced paths, and shortest paths (or interval) transit
functions. Finally we point out some open questions and directions for future
work.
