#19123: LatticePoset: add is_vertically_decomposable
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Reporter: jmantysalo | Owner:
Type: enhancement | Status: needs_work
Priority: major | Milestone: sage-6.9
Component: combinatorics | Resolution:
Keywords: | Merged in:
Authors: Jori Mäntysalo | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
u/jmantysalo/vertically_decomposable|
0d472a68c9edf4ddea1404ff0cb6d2f508de55d0
Dependencies: | Stopgaps:
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Comment (by jmantysalo):
Replying to [comment:5 ncohen]:
> > There are of course other options, like having a function (this one,
with an argument?) returning list of "decomposition elements".
>
> +1 to that.
OK. What should be the name of the argument? `certificate`?
`give_me_the_list=True`?
> > How should it be defined on non-connected posets? And I am not sure if
this works with non-bounded posets; I thinked about bounded ones when
writing this.
>
> Hmmm, okay okay... I attempted to write a definition, but indeed for
non-lattices you have 1000 different corner-cases, and th definition would
be a mess.
Except for the 2-element lattice there is one simple definition that
generalizes this:
{{{
any(P.cover_relations_graph().is_cut_vertex(e) for e in P)
}}}
But in any case, it is easy to move this to posets later if we want so.
> > (Btw, this would be nice exercise of (totally unneeded) optimization.
One should not need to look for all edged of Hasse diagram to see that a
poset is indecomposable.)
>
> What do you mean? Your algorithm looks very reliable. I do not see it
waste much.
If the poset has coverings `2 -> 6` and `4 -> 9`, then no element `3..8`
can be a decomposition element. After founding, say, `2 -> 6` we could
check `5 ->`, `4 ->` and so on. But after founding `4 -> 9` we should have
a somewhat complicated stack to skip re-checking biggest covers of `4` and
`5`. I guess that the algorithm would be slower in reality, but I am quite
sure that it would be better in some theoretical meaning.
--
Ticket URL: <http://trac.sagemath.org/ticket/19123#comment:6>
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