Every co-Heyting algebra can be embedded in the co-Heyting algebra of closed sets in a spectral space. (Co-heyting algebras are the order-theoretic duals of Heyting algebras.) Now I am told the following without a proof.
Every finite co-Heyting algebra can be embedded in the co-Heyting algebra of semialgebraic sets in a real algebraic variety.
Why is this true? The paper that claims this has no reference for this fact, and I am lost. (I am looking for an elementary answer since I know virtually nothing about real geometry.)
A finite distributive lattice can be embedded in a finite power set (Birkhoff representation), which is the lattice of semialgebraic sets of a finite set of points.