Suppose we have $\langle K(n)\rangle$ and $\langle K(n-1) \rangle$ for some fixed prime $p$. do we know whether or not $\langle K(n) \rangle \geq \langle K(n-1) \rangle$ or $\langle K(n-1) \rangle \geq \langle K(n) \rangle$?
It seems to me that the first relation is accurate if we somehow restricted ourselves to finite spectra (from Ravenel's "Localization at Certain Periodic Homology Theories"). However, we obviously aren't doing that (right?).
It also seems to me that such a relationship would be incredibly problematic, mainly because we'd have the following. Assume the first relationship held and we restrict ourselves to operating within the distributive sub-lattice of the Bousfield lattice. Then $\langle K(n)\rangle \wedge\langle K(n-1)\rangle=\langle K(n-1)\rangle = \langle 0\rangle$ and this would seem to me to be incredibly problematic. Is this an accurate assessment of the situation, or have I missed something?