Complete Boolean algebras are Bousfield lattices

TL;DR

This paper constructs tensor triangulated categories with Bousfield lattices corresponding to given complete Boolean algebras, revealing new connections between algebraic structures and lattice theory.

Contribution

It introduces a method to realize any complete Boolean algebra as a Bousfield lattice of a tensor triangulated category, expanding understanding of their relationship.

Findings

Any complete Boolean algebra can be realized as a Bousfield lattice.

Constructs an algebraic tensor triangulated category from a complete Heyting algebra.

Provides examples illustrating complex behaviors of Bousfield lattices.

Abstract

Given a complete Heyting algebra we construct an algebraic tensor triangulated category whose Bousfield lattice is the Booleanization of the given Heyting algebra. As a consequence we deduce that any complete Boolean algebra is the Bousfield lattice of some tensor triangulated category. Using the same ideas we then give two further examples illustrating some interesting behaviour of the Bousfield lattice.