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### Maximal matching of a poset

Hi,

I would like to know how to find a maximal matching (defined below) of a given finite poset.

 A matching  M in a poset X is a subset M of X x X such that:
1) if (x, y) is in M, then x is an immediate predecessor of y (x<y and there is no element z such that x<z<y)
2) each element x of X belongs to at most one element in M.


A matching of a poset is the same as a matching of its Hasse Diagram. I thought about using the matching() function for graphs, however, it seems that the Hasse diagram of a poset is a Digraph for Sage and so the matching function does not work.

By maximal matching I mean one with the maximal number of elements.

Any help would be appreciated.