Here lattice means a poset with meets and joins. A lattice is called atomic if every element is a join of atoms. There are a few different ways to define modular for finite lattices: one is that the lattice is graded and the rank function $\rho$ satisfies $\rho(x)+\rho(y)=\rho(x \vee y) + \rho(x \wedge y)$ for all elements $x,y$.

There is a classification of atomic, modular finite lattices which says that every such lattice is a product $L_1 \times \cdots \times L_k$ of lattices $L_i$ of the following forms:

  • two element lattice;
  • "projective line," i.e., rank 2 modular lattice with at least 5 elements;
  • "projective plane," i.e., rank 3 modular lattice capturing the incidence structure of an abstract finite projective plane;
  • the lattice of $\mathbb{F}_q$-subspaces of $\mathbb{F}_q^d$ for some $d\geq 3$ and $q$ a prime power.

[Since non-Desarguesian finite projective planes are likely not classifiable, maybe this does not 100% qualify as a classification, but anyways it is the result I am interested in.]

For example, as I mentioned in a previous MO answer (https://mathoverflow.net/a/362287) this classification appears in a textbook of Cameron. It is also stated on pg. 48 Stanley's lecture notes on hyperplane arrangements (An Introduction to Hyperplane Arrangements). There it is said to be a consequence of the "fundamental theorem of projective geometry."

I looked up the "fundamental theorem of projective geometry" and it looks like it first appears in a 1907 paper of Veblen. But I don't totally understand how it relates to this result in lattice theory, and I guess that maybe at the time of Veblen the relationship between discrete geometry and lattice theory was not entirely understood. So... does this result first appear in a text of (Garrett) Birkhoff? Does anyone know the definitive source?



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