# Distributive sublattices of geometric modular lattices

I found a copy of Lattice Theory (by Birkhoff) in a dusty corner of our library. I just picked it up for fun and seems really interesting. I was mainly interested in geometric modular lattices.

My question is:

Is there a bound on the size of largest distributive sublattice of a modular geometric lattice?

Additionally, I would like to know how to find good books, lecture notes on lattice theory. I would welcome any suggestions about the order of reading it too.

Thanks,

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Incidentally someone posted a suggestions about books on lattices here, although that question is about a different type of lattices. BTW I think it would be better ask these two things as separate questions - one about the bound and one about the book recommendations. –  Martin Sleziak Aug 9 '12 at 6:56
You can find J. B. Nation's Notes on lattice theory at this website: bigcheese.math.sc.edu/~mcnulty/alglatvar –  Martin Sleziak Aug 9 '12 at 6:59
I've posted a follow-up question which only asks about recommendation for texts on lattice theory (=the last part of your question). –  Martin Sleziak Aug 9 '12 at 8:33

The largest distributive lattice of rank $n$ has $2^n$ elements, so the largest distributive sublattice of any lattice of rank $n$ has size at most $2^n$. For any geometric lattice of rank $n$ this bound can be achieved by taking the sublattice generated by $n$ atoms forming a basis for the underlying matroid.