I have three questions on lattices:
-Is there a way to obtain the minimal number of generators of a lattices with sage?
-Is there a way to obtain the lattice of all subspaces of a vector space over a finite field with q elements in sage?
-Is there a quick way to obtain all distributive lattices on n points in sage (that is, without filtering them from the set of all posets on n points).
Here is a first attempt to answer your questions.
1) The comment should be addressed in order to give an instructive answer.
2) Assuming you want to consider only _linear_ subspace, the following code does the thing:
sage: dim = 2
sage: FF = FiniteField(3)
sage: VS = VectorSpace(FF,2)
sage: SS = {}
sage: SS[0] = set([VS.subspace([])])
sage: for dim in range(1,dim+1):
....: new_subspaces = set()
....: for v in VS:
....: for ss in SS[dim-1]:
....: new_ss = ss + VS.subspace([v])
....: if new_ss.dimension() == dim:
....: new_subspaces.add(new_ss)
....: SS[dim] = new_subspaces
....:
sage: ground_set = reduce(lambda x,y:x.union(y),SS.values())
sage: the_lattice = LatticePoset((ground_set,lambda x,y: x.is_subspace(y)))
3) See https://ask.sagemath.org/question/389... and https://ask.sagemath.org/question/389...
Asked: 2020-01-05 20:01:53 +0100
Seen: 482 times
Last updated: Jan 08 '20
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Could you clarify which lattice you mean? A lattice can be a subgroup of the additive group
R^nwhich is isomorphic toZ^nor a poset with a join and meet operations.Thanks, yes it is about lattices in the sense of posets.