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Saturated chains

asked 2018-08-31 19:16:47 -0600

Stiven gravatar image

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?

rburing gravatar imagerburing ( 2018-09-02 09:26:07 -0600 )edit
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slelievre gravatar imageslelievre ( 2018-09-03 04:41:38 -0600 )edit

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

Stiven gravatar imageStiven ( 2018-09-03 08:08:34 -0600 )edit

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

rburing gravatar imagerburing ( 2018-09-03 09:38:39 -0600 )edit

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answered 2018-09-03 03:56:59 -0600

FrédéricC gravatar image

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

rburing gravatar imagerburing ( 2018-09-03 09:30:28 -0600 )edit

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Asked: 2018-08-31 19:16:47 -0600

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Last updated: Sep 03