# Hasse diagrams

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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.