# Distributive lattice of subspaces

Let $$V$$ be a finite dimensional vector space. Let $$\Lambda$$ be a collection of subspaces of $$V$$ such that, if $$X$$ and $$Y$$ are in $$\Lambda$$, then $$X\cap Y$$ and $$X+Y$$ are in $$\Lambda$$. This makes $$\Lambda$$ into a lattice; we impose that $$\Lambda$$ is a distributive lattice, meaning that $$X \cap (Y+Z) = (X \cap Y) + (X \cap Z) \qquad \forall\ X,\ Y,\ Z \in \Lambda$$ I want a reference for the following statement:

There is a basis $$B$$ for $$V$$ such that, for every $$X \in \Lambda$$, the intersection $$B \cap X$$ is a basis of $$X$$.

The only reference I have been able to find are these notes on Koszul algebras, where it is described as "a standard result in lattice theory". Those notes cite Gratzer, "Lattice theory", 1971, Section 2.7, Theorem 19 but, in fact, Theorem 19 of Section 2.7 in Gratzer does not mention vector spaces at all.

• Mariano's answer to math.stackexchange.com/questions/63493/… suggests another possible reference Commented Jul 29, 2022 at 16:25
• It's proved in Proposition 7.1 in Mariano's reference Quadratic Algebras by Polishchuk and Positselski Commented Jul 29, 2022 at 16:33