Perils of Modern Living: Blogging Edition

With all of its benefits, there are many difficulties of life that are unique to civilization: traffic, taxes, pollution. Thanks to blogging, I’ve experienced a new one: having your WordPress site hacked. The hosting company shut down the site because it was receiving unusual traffic. Upon further investigation, it turned out the site had been hacked. (Presumably the traffic was from the site being made part of a botnet.) It should now be fixed.

Classification of Finite Simple Semigroups and Moufang Loops

I had a question that I was going to ask on Math Overflow, but after some research I managed to find the answer.

Finite simple groups have a complete classification. I was wondering if there were any weakenings of the axioms of group that also allowed a complete classification of the simple objects. (Here, I mean no nontrivial quotients.) Surprisingly, there’s a classification for semigroups. In the theory of semigropus the term “Simple& is used for a weaker notion. Semigroups with no nontrivial quotients are known as “congruence-free”. The classification of finite congruence-free semigroups splits into two cases: for semigroups with a zero (an element 0 such that 0x = 0) there’s an explicit construction, while a congruence-free semigroup without a zero must be a simple group.

Another direction to generalize is weaken the form of associativity. The most-studied weakening is the Moufang property, which includes the octonions as a non-trivial example. Here, the complete classification is also known: a finite simple Moufang loop is either a group or a Paige loop, which is a non-associative construction closely related to the octonions, but defined over a finite field. It’s interesting that in this case, the one non-associative family resembles simple groups of Lie type, in that it’s parameterized by the finite fields. This classification relies non-trivially on the classification of simple groups, in that the explicit classification is used to rule out any other non-associative examples.

The paper Octonions, simple Moufang loops and triality by Gábor Nagy and Petr Vojtechovský, explains Moufang loops, and how the classification of non-associative Moufang Loops reduces to a question about finite simple groups.

Stacks Project

I’ve been trying to learn about stacks, something that is much easier in the Internet age. The Stacks Project is a collaborative textbook that introduces the subject from the ground up, including all of the machinery necessary. The book is already up to 3000(!) pages.

Geometric Logic at Sea

Doing a search for the definition of geometric logic, I have discovered that it’s mentioned in the movie The Caine Mutiny, by the notorious character of Captain Queeg:

Ahh, but the strawberries that’s… that’s where I had them. They laughed at me and made jokes but I proved beyond the shadow of a doubt and with… geometric logic… that a duplicate key to the wardroom icebox DID exist, and I’d have produced that key if they hadn’t of pulled the Caine out of action.

The upside of the technique is that it allows you to deduce the existence of real-world objects such as keys. The downside is that it drives you insane.

(More on the context of the quote here.)

Commutativity Theorems for Rings

MathJax doesn’t work with RSS readers, so when I have some more time I will look into using a plug-in instead. Until then, sorry for filling up your RSS feed with dollar signs.

Theo Raedschelders has written a nice sketch of Herstein’s commutativity theorem for rings. It is a generalization of Wedderburn’s theorem that a finite division ring must be a field. The theorem states that if for every pair of elements $a$ and $b$ there exists an $n > 1$ (which can depend on $a$ and $b$) such that
(ab – ba)^n = ab – ba,
then the ring is commutative. What’s surprising about the proof is its indirectness. The proof requires essentially all of Nathan Jacobson’s structure theory for rings.

PlanetMath has a nice summary of known conditions that imply commutativity.


As of a few hours ago, all I know about quandles was that they had something to do with knots. Since in our modern connected age ignorance lasts only as long as you want it to, I decided to find out more. I found this slide deck by Bob McGrail explains both the definition of quandle and the motivation in knot theory in a measly six slides. (The rest of the talk is how to efficiently compute quandles.)

Sam Nelson has a nice introduction to the general area of universal-algebraic invariants from the Notices, called Revolution in Knot Theory. Nelson also discusses the emergence of virtual knots.

Experimenting with MathJax

I’m experimenting with MathJax to include mathematics on the blog. Part of the incredible slowness in which I write posts is doing everything by hand in HTML. For example, I have a future post for the Horn clause series that I keep not finishing because I get tired of writing the subscript tags. MathJax is the display engine used for Math Overflow and Stack Exchange, which allows you to use tex commands to represent formulas.

Here’s a sample of tex-encoded math formulas that I copied from the MathJax site.

When $a \ne 0$, there are two solutions to \(ax^2 + bx + c = 0\) and they are $$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$

If you actually view the source for the file in your browser, you’ll see the raw tex commands in the post. MathJax uses CSS to render it on the fly. I’m actually surprised this is possible, but apparently it is.

Hilbert’s Foundations of Geometry

Project Gutenberg has David Hilbert’s Foundations of Geometry available. It is a translation of Hilbert’s Grundlagen der Geometrie, which is famous as the first modern axiomization of Euclidean geometry. The difference between Hilbert’s approach and that of Euclid is that Hilbert fills in all of the fiddly little details required to meet modern standards of rigor.

The book is elementary, and (as translated by Townsend) is a pleasant read. Much of the book centers around constructing the field of real numbers in terms of the axiomized geometrical constructions. This in turn allows Hilbert to show that the set of axioms is complete. The topic leads naturally to one of the main themes of research in plane geometry in the early part of the last century, which is to consider different algebraic objects and how they can serve as coordinates for different notions of affine or projective planes. The reals can be replaced with an arbitrary division ring, for example. For a projective plane, the most general object is a planar ternary ring, with has a ternary operation that serves as a hybrid of addition and multiplication. Determining the projective planes with a finite number of points is still an open question.