Simplicial-like Identities for The Paths and The Regular Paths on Discrete Sets

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.