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Let $L$ be a finite lattice and $A=KL$ the incidence algebra of $L$. It should be true that $L$ is modular if and only if the algebra $KL$ is quadratic (since being modular is equivalent to having no ...

I noted a while ago that there is a nice homological formulation using incidence algebra of the Union-closed sets conjecture (https://en.wikipedia.org/wiki/Union-closed_sets_conjecture). It might just ...