Warning: This page is tentative and may contain errors.
In other words, a Hasse diagram is a directed graph in which for each edge there is no other path from to . There are no intermediate edges.
In particular, given a proset , its Hasse diagram is obtained by “forgetting all composite morphisms”. The proset may then be recovered as the free poset on that Hasse diagram.
More formally, there is a forgetful functor
H: Ord \to Hasse,
where is the category of preordered sets and is the category of Hasse diagrams, that forgets composite morphisms.
The corresponding free functor
allows us to identify a Hasse diagram with each proset.