Inhabitants of interesting subsets of the Bousfield lattice

Abstract

The set of Bousfield classes has some important subsets such as the distributive lattice $\mathbf{DL}$ of all classes $⟨E⟩$ 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 $⟨H{\mathbb{F}}_p⟩\in \mathbf{DL}\setminus \mathbf{cBA}$.