# 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.

Here is a link to the PDF version of my notes.

Short Attention Span Math Seminars—or SASMS—is an event held by the University of Waterloo’s Pure Math Club every four months. It’s a fun little evening where students can sign up to give short 30 minute talks on topics to interest them, and it’s a fun way to practice speaking. I was club president for probably like a billion years, and I’ve signed up to give a talk every term since before my father was born, so of course I talked this term. This is probably the last talk I’m going to give, so I wanted to do a good job for once. I think it went over well.

The talk is about lattices, of course, because lattices are basically my favourite things that aren’t matroids.
A lattice is **distributive** if its meet operation distributes over its join, or equivalently vice versa.
It turns out that there is a nice excluded-subobject theorem characterizing distributivity—akin to Kuratowski’s theorem on planarity—and it “factors” nicely into two halves as far as lattice theory is concerned.

I think I’ll spend the rest of this post explaining the story behind why I’m gave this talk.

Last fall, I took a course called “Introduction to Substructural Logics”. This was an undergraduate topics course in philosophy, so it counts as an elective on my transcript, but it was taught by a hardcore logician and was very heavily mathematical in nature. Basically a perfect course for someone like me who is lazy and can’t stand doing things that aren’t math. There were at most twelve people taking it, most of them probably upper-year philosophy students with logic leanings. Me and my good friend Sean Harrap, a mathematician and a computer scientist respectively, were also students.

It proceeded slowly enough at first, with many breaks to talk about the history or some connections to philosophical ideas, but it walked and talked like an easy math course. The grading scheme was to be based on the class’ performance on three assignments, with the explicit expectation that people should try to complete as many questions as they can and the grading curve would be decided holistically after the fact. Having enough time and interest on my hands, Sean and I decided to try to answer every question. For the first two assignments, this was not very hard or time-consuming, as the exercises were fairly simple.

By the time the third assignment rolled around, however, we had begun to cover algebraic semantics, and while the lectures still proceeded at a reasonable pace, the assignment pulled out all the stops. One of the questions was to prove that a bounded lattice was a Boolean algebra iff every prime filter was an ultrafilter—you may recall I squeezed a series of two blog posts out of this problem—and it was assigned to us as casually as anything. Another question was to prove the previously mentioned excluded subobject result for lattice distributivity, which an exercise I would not recommend for even the most intense training regimens.

I managed to walk out of the course with a hundo, so all’s well that ends well, but I worry the marking scheme was a little too adversarial and might have left some of the less algebraically inclined students high and dry.

In any case, this was a tale I’d recounted many times since, and at one point, as a joke, my partner in substructural crime Sean Harrap suggested I give a talk about this course since I couldn’t stop yammering on about it. So I put my money where my mouth was and set to work digesting this proof until it fit into 30 minutes. Even then it’s a bit of a stretch, but at the very least all the necessary ingredients are there.