An Adjunction Between Boolean Algebras and a Subcategory of Stone Algebras
Inigo Incer
TL;DR
The paper establishes a categorical bridge between Boolean algebras and augmented Stone algebras by introducing the contract algebra functor $\mathbf{C}: \mathbf{Bool}\to \mathbf{augStone}$ and the closure functor $\mathrm{Clos}: \mathbf{augStone}\to \mathbf{Bool}$. It defines augmented Stone algebras via a distinguished closure element $e$ with $e \to x = \mathrm{Clos}(x)$ and shows how $\mathrm{Clos}$ extracts the Boolean algebra of closed elements, while $\mathbf{C}$ furnishes augmented Stone structures from Boolean algebras. The main contribution is a natural bijection between $\hom_{\mathbf{Bool}}(B,\mathrm{Clos}(S))$ and $\hom_{\mathbf{augStone}}(\mathbf{C}(B),S)$, realized by explicit maps, which demonstrates that $\mathbf{C}$ is left adjoint to $\mathrm{Clos}$. This adjunction provides a precise mechanism for transferring structure between Boolean algebras and a subcategory of augmented Stone algebras, linking propositional-like logics with lattice-theoretic semantics.
Abstract
We consider Stone algebras with a distinguished element $e$ satisfying the identity $e \to x = \neg \neg x$ for all elements $x$ of the algebra. We provide an adjunction between the category of such algebras and that of Boolean algebras.