  Simplicial identities play an important and fundamental role in simplicial
homotopy theory. On the other hand, the study of the paths and the regular
paths on discrete sets is the foundation for the path-homology theory of
digraphs. In this paper, by investigating some weighted face maps and weighted
co-face maps on the space of the paths as well as the space of the regular
paths, we prove some simplicial-like identities for the paths and the regular
paths on discrete sets.
