Hi Anne,
> Isn't the following a big bug in posets in sage:
>
> sage: P = Poset(([1,2,3,4],[[1,4],[2,3],[3,4]]), facade = True)
> sage: P.is_graded()
> False
>
> The definition of grading is that all maximal chains of every interval
> have the same length. This is given in the above poset, but sage says
> it is not graded.
The code seems to assume that the poset has a least element. It uses
the level_sets() function of sage/graphs/digraph.py to compute the
level sets of the Hasse diagram of the poset; then it assumes that the
0-th level set (i. e., the set of vertices having no predecessor) must
be the 0-th rank of the grading, that the 1-st level set (i. e., the
set of vertices having all their predecessors in the 0-th level set
but not lying therein themselves) must be the 1-th rank, that the 2-nd
level set (i. e., the set of vertices having all their predecessors in
the <=1-st level sets but not lying therein themselves) must be the
2-nd rank, etc. In the case of your P, the 0-th level set is {1,2},
and then we get a rank mismatch at vertex 4.
> How should this be fixed?
I am wondering about that too. My first idea would be Gaussian
elimination, with each covering relation (P.cover_relations()) giving
an equation of the form a_i - a_j = 1. Not sure this is anywhere near
optimal, though. The nice thing about systems of linear equations with
every equation containing only 2 nonzero coefficients is that, if you
perform Gaussian elimination on such a system, it retains this
property (that every equation has only 2 nonzero coefficients) all the
way through (this is even true for Gröbner basis, keyword "binomial
basis" IIRC), so it is a very sparse-matrix problem, but I don't know
whether Sage has any code optimized for this case.
Best regards, and sorry if this was all known already,
Darij
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