nLab Hasse diagram

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Idea

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

Definition

A Hasse diagram $H$ 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 $x\to y$ there is no other path from $x$ to $y$. There are no intermediate edges.

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

More formally, there is a forgetful functor

$$H: Ord \to Hasse,$$

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

The corresponding free functor

$$F:Hasse\to Ord$$

allows us to identify a Hasse diagram with each proset.

References

• See also Hasse n-graph
• Wikipedia