As is well-known, the lattice of submodules of a module is modular. What I did not know is that the converse is not true, and that lattices of submodules must satisfy a stronger property, the arguesian law.

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.)

Whenever you have a lattice of commuting equivalence relations, as is the case for lattices of submodules, or more generally lattices of submodels of a model of a Mal’cev algebraic theory, you have an infinite list of modular-law type identities of which the arguesian law is but one. This isn’t an area of expertise of mine, but you can learn more from this article courtesy of Google books, and also in Freyd and Scedrov’s Categories, Allegories, section 2.158. Freyd and Scedrov indicate that the theory of lattices satisfying all of these modular-type identities is decidable, and to me it smells very similar to the calculus developed by Rota and his collaborators, but that there are close connections between these works should be treated for now as a hunch.

Thanks for the references. That’s surprising, because the theory of modular lattices is undecidable. I was wondering if the theory of sublattices of modules (or in general the theory of congruence lattices of congruence modular varieties that you mention) is decidable, and I was thinking about asking on Math Overflow. I should check your references first.

I have the impression that the complete list of the identities satisfied in these cases is not yet known.