nLab
Hasse n-graph

Warning: This page is tentative.

Idea

Given a finite n-poset $C$, its Hasse n-graph encodes the minimal amount of information necessary to reproduce the ordering relation.

Definition

Given an n-poset $C$, a Hasse $n$-graph $H$ is a directed n-graph whose Hasse quiver $F(H)$ is equivalent to $C$.

As with Hasse diagram, I think that declaring it to be smallest in unnecessary; the fact that it is merely an $n$-graph and not a $n$-category will do this. But this should be an $n$-quiver, yes? —Toby

Eric: What should be an n-quiver? I thought I’d stick the “smallest” in there because if $x<y$, $y<z$ then the morphisms $x\to y$, $y\to z$, and $x\to z$ are in the quiver, but only $x\to y$ and $y\to z$ need to be in the Hasse diagram. However, we could have a graph including $x\to z$, 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.