Quantum geometry of Boolean algebras and de Morgan duality

Abstract

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