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.


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Just to not leave this open, a proof can be found in Proposition 7.1 of Chapter 1 of Quadratic Algebras by Alexander Polishchuk and Leonid Positselski as alluded to by Mariano on MSE https://math.stackexchange.com/questions/63493/non-distributivity-of-subspaces.


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