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Given a finite n-poset , its Hasse n-graph encodes the minimal amount of information necessary to reproduce the ordering relation.
As with Hasse diagram, I think that declaring it to be smallest in unnecessary; the fact that it is merely an -graph and not a -category will do this. But this should be an -quiver, yes? —Toby
Eric: What should be an n-quiver? I thought I’d stick the “smallest” in there because if , then the morphisms , , and are in the quiver, but only and need to be in the Hasse diagram. However, we could have a graph including , but this would not be a Hasse diagram. Maybe I’m confused.
Toby: It would not be a Hasse diagram, but also its quiver would not be a poset, so that's taken care of.
However, the basic idea was wrong anyway, as shown by your counterxample here.