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.

翻译：传递函数最初被引入作为无向结构中介关系的模型。本文提出有向传递函数，作为偏序集和有向图等有向结构上的有向类比。我们首先证明偏序集中的介关系可以通过一组简单的一阶公理来表达。对于具有自然偏序的图，特别是森林、树和红树林，也可以获得类似的表征。放宽无环性条件后，可将著名的几何传递函数推广到有向结构。此外，我们讨论了重要传递函数的有向类比的一些性质，包括全路径、诱导路径和最短路径（或区间）传递函数。最后，我们指出一些未解决的问题和未来工作的方向。