Revision history [back]

Question: Is there a quick way to obtain all atomic lattices with $n$ elements and $k$ atoms for given $n,k$?

Here atomic means that every element is a join of atoms in the lattice.

I think such a lattice has at most $2^k$ elements and thus there shoud be a quick method.

I want to use a program for sage to obtain a large list of atomic lattices but the brute force way just works for $n \leq 12$.

Thanks for any help!

Question: Is there a quick way to obtain all atomic lattices with $n$ elements and $k$ atoms for given $n,k$?

Here atomic means that every element is a join of atoms in the lattice, where atoms are the elements that cover the minimal element of the lattice.

I think such a lattice has at most $2^k$ elements and thus there shoud be a quick method.

I want to use a program for sage to obtain a large list of atomic lattices but the brute force way just works for $n \leq 12$.

Thanks for any help!

n=8
P=Posets(n-2)
P=[p.with_bounds() for p in P]
L=[p for p in P if p.is_lattice() and LatticePoset(p).is_atomic()]
display(len(L))
display(L[2])