Commutativity conditions for groups arising from acyclic directed graphs and posets

Several groups are associated naturally with acyclic directed graphs and in particular with partially ordered sets. It is investigated when these groups are abelian. Those acyclic directed graphs where the groups are abelian are characterized as in-Eulerian, out-Eulerian, Eulerian and strongly Eulerian. Here strongly Eulerian means that for any two elements the parity of their common out-neighbors is equal to the parity of their common in-neighbors. We also give an upper bound for the order of the mod 2 raising operator which is the most prominent generator of the groups.