More lattice stuff
I’ve continued reading about lattice theory. It’s exactly what I need to stimulate an interest in my algebra class.
One interesting fact I came across was that a lattice is distributive iff it doesn’t contain a diamond or pentagon as a sublattice. That’s interesting not only because it is a totally unexpected condition for sufficiency, but also in graph theory, there’s a similar condition on the planarity of graphs. I wonder if there’s any significant connection between lattices and graph theory, via directed graphs maybe?
Another interesting fact: 
since 
is a distributive lattice (unfortunately, this equality has to be proven to show that it is in fact a distributive lattice), and since this is true, so is the formula 
(this one is automatic, since the first one shows that it is a distributive lattice).
Possibly relevant posts:
- Practical Graph Theory (2/18/2003)
- Lattice Problem (2/16/2005)
- spectral graph theory and two CS applications (5/4/2008)