# Saturated chains

I know there is a command to take all maximal chains and all cover relations of a poset. I am working with a ranked poset and I am wondering if there is a command for all saturated chains in a poset. I appreciate any help given.

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This is not my field so I can ask a silly question: Is every saturated chain a subchain of a maximal chain?

( 2018-09-02 16:26:07 +0200 )edit
1
( 2018-09-03 11:41:38 +0200 )edit

Yes. Every saturated chain is a maximal chain of a subposet.

( 2018-09-03 15:08:34 +0200 )edit

My naive idea was to take all subchains of all maximal chains. This works, but it is inefficient because you get duplicates.

( 2018-09-03 16:38:39 +0200 )edit

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Like that

sage: p = posets.DiamondPoset(6)
sage: h = p.hasse_diagram()
sage: [h.all_paths(x,y) for x in h for y in p.principal_order_filter(x)]
[[[0]],
[[0, 1]],
[[0, 2]],
[[0, 3]],
[[0, 4]],
[[0, 1, 5], [0, 2, 5], [0, 3, 5], [0, 4, 5]],
[[1]],
[[1, 5]],
[[2]],
[[2, 5]],
[[3]],
[[3, 5]],
[[4]],
[[4, 5]],
[[5]]]

more

I would wrap the last command in sum(..., []) to get a single list of chains.

( 2018-09-03 16:30:28 +0200 )edit