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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/38938/finding-certain-posets-in-sage/ and https://ask.sagemath.org/question/38936/obtaining-lattices-quickly-in-sage/

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)))x.is_subspace(y)))