nLab
Hasse diagram

Hasse diagrams

Warning: This page is tentative and may contain errors.

Idea

Given a locally finite partially ordered set CC, its Hasse diagram encodes the minimal amount of information necessary to reproduce the ordering relation.

Definition

A Hasse diagram HH is a directed graph (or quiver) such that the adjacency relation equals the covering relation.

In other words, a Hasse diagram is a directed graph in which for each edge xyx\to y there is no other path from xx to yy. There are no intermediate edges.

In particular, given a proset CC, its Hasse diagram H(C)H(C) is obtained by “forgetting all composite morphisms”. The proset CC may then be recovered as the free poset on that Hasse diagram.

More formally, there is a forgetful functor

H:OrdHasse,H: Ord \to Hasse,

where OrdOrd is the category of preordered sets and HasseHasse is the category of Hasse diagrams, that forgets composite morphisms.

The corresponding free functor

F:HasseOrdF:Hasse\to Ord

allows us to identify a Hasse diagram with each proset.

References

Revised on February 13, 2011 20:02:54 by Toby Bartels (75.88.68.70)