For given two digraphs, we can construct a larger digraph through join. The digraphs that make up the join are called factors of In this paper, give necessary and sufficient condition function on determined by discrete Morse functions is function. Moreover, further prove theory when satisfy certain conditions.

<abstract><p>We prove a version of the fundamental theorems Morse theory in setting finite partially ordered sets. By using these results we extend Forman's discrete to more general cell complexes and derive Morse-Pitcher inequalities that context.</p></abstract>