It is know that for any $\alpha$well generated tensor triangulated category $\mathcal{T}$ the collection of Bousfield classes forms a set, furthermore this set is a complete lattice, denoted by $\mathscr{B}(\mathcal{T})$, and this lattice contains a subposet $\mathscr{D}(\mathcal{T})$ consisting of the Bousfield idempotents, this is a lattice, in fact is frame (locale). My question is the following: Is there some knowledge concerning (partial) distributive laws in the lattice $\mathscr{B}(\mathcal{T})$, for example, upper continuity, that is, the following identity: $a\wedge(\bigvee X)=\bigvee \left\{a\wedge x x\in X\right\}$ holds for every directed $X\subseteq\mathscr{B}(\mathcal{T})$ and $a\in\mathscr{B}(\mathcal{T})$, or modular law?
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$\begingroup$ Could you please give a reference for the frame of Bousfield idempotents? $\endgroup$ – მამუკა ჯიბლაძე Apr 15 '15 at 4:45

1$\begingroup$ Yes, this paper is related with that arxiv.org/pdf/1105.1799.pdf and arxiv.org/pdf/math/9801103.pdf $\endgroup$ – Angel Zaldívar Apr 15 '15 at 4:47