#19197: LatticePoset: add breadth()
-------------------------------------+-------------------------------------
Reporter: jmantysalo | Owner:
Type: enhancement | Status: new
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/latticeposet__add_breadth__|
8d81130c3ea697cbfe3ce178dd5a7fac052e78d3
Dependencies: | Stopgaps:
-------------------------------------+-------------------------------------
Comment (by jmantysalo):
Here is, if I thinked this right, the smallest lattice with breadth 4:
{{{
n = 4
l = [[0,i] for i in range(1,n+1)]
l += [[i, i+n] for i in range(1,n+1)]
l += [[i, i+n+1] for i in range(1,n)]
l += [[n, n+1]]
l += [[i, i+n] for i in range(n+1, 2*n+1)]
l += [[i, i+n+1] for i in range(n+1,n*2)]
l += [[n*2, n*2+1]]
l += [[i, n*3+1] for i in range(2*n+1, 3*n+1)]
L = Poset(( [], l)).completion_by_cuts().canonical_label()
}}}
This is not easily generalized. `n=5` will not make a lattice of breadth
5.
However, can this be used as a part of test for a lattice to have breadht
4? If br(L)=4, there must be an antichain `A` of `4` elements with join
`j`. Element `j` must cover at least four element. They can not be the set
`A`. They can not directly cover `A`, because it is just impossible to
make the lattice so that three element from `A` would not have meet `j`.
So to check take one element a time and try to put in as a `j` in this
structure. If the element covers less than four element, it is not right
one. If it does, take all four-element subsets of the lower covers of
`j`-candidate. For those... somehow continue a-kind-of backtracking.
We can remove all doubly irreducible elements, as removing thme won't
change the breadth, except from 2 to 1 in a diamond-like stucture.
--
Ticket URL: <http://trac.sagemath.org/ticket/19197#comment:10>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
--
You received this message because you are subscribed to the Google Groups
"sage-trac" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To post to this group, send email to [email protected].
Visit this group at http://groups.google.com/group/sage-trac.
For more options, visit https://groups.google.com/d/optout.
