I’ve been working a little with an old friend, modal logic, this weekend. This digressed into a study of lattices and Boolean algebras, so I thought I would write up some of my observations for archival. Also, I just like to say “lattice”.
If you know anything about lattices, you will probably find this stuff trivial. If you don’t know anything about lattices, a free and much better introduction can be found in A Course in Universal Algebra.
Definition: A distributive lattice is a lattice which satisfies the distributive laws:
Actually, it can be shown that a lattice satisfies D1 iff it satisfies D2.
Definition: A lattice L is said to be modular if the modular law holds:
Lemma: The modular law for lattices is equivalent to the identity
Assume that the modular law holds and consider the expression . Since we can apply the modular law to infer that .
Now, assume that the identity holds and that for some . Then so
where the identity was used in . Thus, the modular law holds.
Theorem: Every distributive lattice is modular.
Assume that for some where is a distributive lattice. Then so
where we have made use of D2. The desired result now follows from the lemma.