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Idea

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