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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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    I've posted a [follow-up question](http://math.stackexchange.com/questions/180578/introductory-text-for-lattice-theory) which only asks about recommendation for texts on lattice theory (=the last part of your question).2012-08-09

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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.

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Richard Stanley's book Enumerative Combinatorics is highly recommended.