nLab
Hasse n-graph

Warning: This page is tentative.

Idea

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

Definition

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

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

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