Inhabitants of interesting subsets of the Bousfield lattice
\textit{A. K. Bousfield} [Comment. Math. Helv. 54, 368--377 (1979; Zbl 0421.55002)] introduced an equivalence relation on the stable homotopy category by declaring two spectra to be equivalent when they determine the same classes of acyclic spectra. This equivalence relation has also become known as Bousfield equivalence. Its class of equivalence classes \(\mathbf{B}\) forms a set, and inclusion of classes of acyclic spectra induces the structure of a lattice on \(\mathbf{B}\). Bousfield [loc. cit.] and \textit{M. Hovey} and \textit{J. H. Palmieri} [Contemp. Math. 239, 175--196 (1999; Zbl 0947.55014)] introduced certain subsets of \(\mathbf{BA} \subseteq \mathbf{cBA}\subseteq \mathbf{DL} \subseteq \mathbf{B}\) of the Bousfield lattice. For example, \( \mathbf{DL}\) is the distributive lattice of all Bousfield classes \(\langle X \rangle\) with the property \(\langle X \rangle \wedge \langle X \rangle = \langle X \rangle\) and \(\mathbf{cBA}\) is the complete Boolean algebra of closed classes. Working \(p\)-locally with respect to a prime \(p\), the paper under review provides lower bounds for the size of \( \mathbf{DL}\setminus \mathbf{cBA}\) and \( \mathbf{cBA}\setminus \mathbf{BA}\). As an application, the authors show that the Bousfield class of the Eilenberg--Mac Lane spectrum \(H\mathbb F_p\) lies in \( \mathbf{DL}\setminus \mathbf{cBA}\).
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