Quantum geometry of Boolean algebras and de Morgan duality

We take a fresh look at the geometrization of logic using recently developed tools “quantum Riemannian geometry” applied in digital case over field $\mathbb{F}\_2={0,1}$, extending de Morgan duality to this context differential forms and connections. The $1$-forms correspond graphs exterior derivative subset amounts arrows that cross between set its complement. line graph $0-1-2$ has non-flat but Ricci flat quantum geometry. previously known four geometries on triangle graph, which one is curved, are revisited terms left-invariant differentials, as dual Hopf algebra, group algebra $\mathbb{Z}\_3$. For square, we find moduli geometries, all flat, while for an $n$-gon with $n>4$ unique one, again flat. also propose extension general algebras differentials $\mathbb{F}\_2$.