# Bousfield's distributive lattice DL and non-ring spectra

Bousfield, in his paper "The Boolean algebra of spectra" (Comm Math Helv 54, 368–377 (1979), https://doi.org/10.1007/BF02566281), defined $$\mathbf{DL}$$, a sublattice of the Bousfield lattice, to consist of all Bousfield classes $$\langle X \rangle$$ such that $$\langle X \rangle \wedge \langle X \rangle = \langle X \rangle$$. He pointed that if $$X$$ is a wedge of ring spectra, then $$\langle X \rangle \in \mathbf{DL}$$. Are there classes in $$\mathbf{DL}$$ which are known not to be Bousfield equivalent to a wedge of ring spectra? If the telescope conjecture fails, then that would yield examples. Are there others?

My paper A combinatorial model for the known Bousfield classes defines an complete ordered semiring $$\mathcal{A}$$ and a homomorphism from $$\mathcal{A}$$ to the Bousfield lattice mod the telescope conjecture, whose image contains most of the Bousfield classes that have been named and studied. In $$\mathcal{A}$$, all elements satisfying $$x\wedge x=x$$ correspond to wedges of unital ring spectra. (In fact, they all correspond to unital ring spectra, except for the case of $$\bigvee_{i\in U}K(i)$$, which is not obviously a unital ring if $$U$$ is infinite.) Although the main results are stated in a quotient of the Bousfield lattice where the telescope conjecture is forced to be true, many intermediate results are stated in the Bousfield lattice itself. I am therefore fairly confident that the literature does not contain any unconditional counterexamples for your question.