Abstract: In a 2003 paper, Herzog and Hibi study the Hibi ideal of a poset and describe
its resolution in detail. Furthermore, they use this notion to classify Cohen-Macaulay bipartite graphs. In 2007 work, Carra'Ferro and Farrarello demonstrate a construction associating a bipartite undirected graph $G$ to a directed graph $D$, and re-cast the aforementioned classification in terms of the directed graph $D$. In particular, they
show that $G$ is Cohen-Macaulay if and only if $D$ is acyclic and transitive. We study how and to what extent this result generalizes to to directed hypergraphs. Though we do not achieve a characterization as in the case of directed graphs, we demonstrate a
class of directed hypergraphs and a sufficient condition for the associated undirected hypergraph to be Cohen-Macaulay.
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