# To whom is the classification of atomic, modular finite lattices due?

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?

• Frink attributes it to Birkhoff Garrett Birkhoff, Lattice theory, Amer. Math. Soc. Colloquium Publications, vol. 25, 1940. See ams.org/journals/tran/1946-060-00/S0002-9947-1946-0018635-9/… although he is talking complemented Sep 23 at 2:39
• @BenjaminSteinberg: Thanks, that's a nice reference! IIRC for finite modular lattices, "atomic" and "relatively complemented" are the same, so probably it is not a big distinction... Sep 23 at 2:44
• Maybe jstor.org/stable/1968656 is relevant Sep 23 at 2:46