Quantum geometry of Boolean algebras and de Morgan duality

TL;DR

This paper explores quantum Riemannian geometry applied to Boolean algebras over f_2, extending de Morgan duality to differential forms and connections, revealing geometric structures on graphs and algebraic objects.

Contribution

It introduces a novel quantum geometric framework for Boolean algebras over f_2, extending de Morgan duality and analyzing geometric properties of graphs and Hopf algebras.

Findings

The line graph 0-1-2 has a non-flat Ricci-flat quantum geometry.

The triangle graph admits four quantum geometries, one curved.

The square graph has four flat quantum geometries, and larger polygons have a unique flat geometry.

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 $\Bbb F_2=\{0,1\}$, 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 $\Bbb Z_3$. 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 $\Bbb F_2$.