Inhabitants of interesting subsets of the Bousfield lattice

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