• ## Prime Filters in Distributive Lattices III

Recall from PFDL II, I gave an interesting characterization of Boolean algebras among distributive lattices, using a technique from formal logic. Today I’d like to share some final musings on the topic, specifically in the form of a counterexample to a weakening of the hypotheses...

• ## Review of This 'Blog

This ‘blog has a lot of lattices in it. ¶ Some of them are kinda hard to imagine, like countable boolean algebras, where you have to use your brain and count or something. Some of them are really hard to imagine, like the free modular lattice on three generators, which for some strange reason is finite and proving that requires over a half hour of computations...

• ## An excluded subobject theorem for nondistributive lattices

I gave a talk yesterday at the PMC’s SASMS. I typeset some notes, and I thought I would share them because I drew pictures. This typeset version has more details and less visual intuition than what I presented on the blackboard, which is why the whole thing fit into a half hour. Find a link to the notes and some flavour text under the cut...

• ## sl(2) for Combinatorialists

There is a long and terrific story to tell about Lie theory, and I wish I could do it justice, but there’s far too much to say in a single post. What I have today is merely one application of one Lie algebraic idea, which ends up being a useful theoretical and practical tool in enumerative combinatorics...

• ## Graph Homomorphisms and Cores

Today I’d like to talk about an open problem I’ve been interested in for the past couple of years. It’s a very hard problem, in that there are easier special cases that are famously unapproachable, but it makes for some rather pretty algebra, living at the intersection of graph theory and category theory. If you ask me, it’s too good to be false...

• ## A Few Translation Exercises

One of my favourite books is Gödel, Escher, Bach, by Douglas Hofstadter. If you’re patient enough to read it all, I highly recommend it: it’s a great book about mathematics and cognitive science. ¶ One of the main goals of the book is to motivate, and sketch a proof of, Gödel’s first incompleteness theorem. At one point, he provides some exercises in transcribing number theoretical statements in a specific implementation of Peano arithmetic he calls TNT...