# Inhabitants of interesting subsets of the Bousfield lattice

## TL;DR

This paper explores specific subsets of the Bousfield lattice, providing examples of spectra that belong to the distributive lattice but not to the Boolean algebra, highlighting structural distinctions within the lattice.

## Contribution

It offers explicit examples of spectra in the distributive lattice that are not in the Boolean algebra, including for every prime p, the Bousfield class of Hℱₚ.

## Key findings

- Bousfield classes form important subsets like the distributive lattice and Boolean algebra.
- Examples of spectra in the distributive lattice but not in the Boolean algebra are provided.
- The Bousfield class of Hℱₚ is in the distributive lattice but not in the Boolean algebra.

## Abstract

The set of Bousfield classes has some important subsets such as the distributive lattice $\mathbf{DL}$ of all classes $\langle E\rangle$ which are smash idempotent and the complete Boolean algebra $\mathbf{cBA}$ of closed classes. We provide examples of spectra that are in $\mathbf{DL}$, but not in $\mathbf{cBA}$; in particular, for every prime $p$, the Bousfield class of the Eilenberg-MacLane spectrum $\langle H\mathbb{F}_p\rangle\in\mathbf{DL}{\setminus}\mathbf{cBA}$.