A Dual Categorical Equivalence between DiGraph Posets and DiGraph Lattices

Abstract

It is well known that there is an incidence poset associated with every directed graph. The problem is that this poset doesn’t encode enough information to recreate our directed graph. In his thesis, Crowell solved this problem by considering tripartite posets whose middle elements covers and exactly covered by one element, and which possess a bijection between the maximal elements and the minimal ones. Crowell proved a categorical equivalence between the category DiGraph and the category DiGraph posets [1]. We extend this idea to lattices and we establish a dual categorical equivalence between the categories DiGraph Posets and DiGraph lattices. This will implies a dual categorical equivalence between the categories DiGraph and DiGraph lattices.