The Arguesian law is a lattice-theoretic analogue of Desargues’ theorem in projective geometry. I read the statement of the theorem several times and I have no intuition about what it means.

There is a kind of converse to this result: a complemented lattice can be embedded into the lattice of submodules of a module if and only if it is arguesian. (I found the result in Gratzer’s book on lattice theory, which is viewable in Google Books.)

]]>The connection between classical topics in algebraic geometry and modern techniques is fascinating.

]]>Project Gutenberg has a book by Schafer on the general theory of nonassociative algebras. Kevin McCrimmon has an unpublished draft of a book on the structure theory of alternative algebras.

I previously linked to an article classifying the simple Moufang loops. The only examples that are not groups are again related to the octonions.

]]>The main source of interest in determinacy is that results suggest that it is the strongest regularity property that a set can have, in that it it tends to imply other nice properties such as Lebesgue measurability. Here is a short proof by Martin that determinacy implies Lebesgue measurability. Justin Palumbo has a nice set of lecture notes that relate determinacy to other regularity properties.

(One nuance is that determinacy for a single set usually doesn’t imply strong regularity properties — the proofs typically require several auxiliary games for a single set. The Martin and Palumbo links use the setting of the axiom of determinacy, which is the axiom that all sets are determined. This is actually false in ZFC: it contradicts the axiom of choice. There are analogous results that hold in ZFC where you keep track of which sets you need to have determined.)

]]>An earlier expository paper, Faltings plus epsilon, Wiles plus epsilon, and the Generalized Fermat Equation, talks about the general question of finding solutions to the “generalized Fermat eqution”, x^{p} + y^{q} = z^{r}.

Linear types are one of those things that I’d always wanted to learn more about. The idea seems somewhat natural &emdash; practically speaking the amount of resources an object uses is part of its signature &emdash; but the details are sufficiently complex that I’ve never quite mastered it. This presentation by Francois Pottier seems like a nice place to start.

Via Lambda the Ultimate.

]]>Simon Willerton has thoughts on PERT charts as copresheaves.

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