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 $x\lt y$, $y\lt z$ 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.

Revised on September 4, 2010 21:50:26
by Toby Bartels
(173.190.156.19)