It is know that for any $\alpha$-well genereted 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 sublattice $\mathscr{D}(\mathcal{T})$ consisting of the bousfield idempotents, this lattice is a 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?