Quantum geometry of Boolean algebras and de Morgan duality

[摘要] 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 F2={0,1}\mathbb{F}_2=\{0,1\}F2​={0,1}, extending de Morgan duality to this context of differential forms and connections. The 111-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 0−1−20-1-20−1−2 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 Z3\mathbb{Z}_3Z3​. For the square, we find a moduli of four quantum Riemannian geometries, all flat, while for an nnn-gon with n44n4 we find a unique one, again flat. We also propose an extension of de Morgan duality to general algebras and differentials over F2\mathbb{F}_2F2​.

[学科分类] 神经科学

[关键词] Logic;noncommutative geometry;digital geometry;quantum gravity;duality;power set;Heyting algebra;poset