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For a spectrum $X$, Bousfield constructs a spectrum (which is only well-defined up to Bousfield equivalence) $aX$, which he shows satisfies some nice properties, like $\langle a^2X\rangle=\langle X\rangle$, and converts arbitrary joins to meets and arbitrary meets to joins. If we restrict ourselves to Bousfield idempotent spectra ($\mathbf{DL}$), or even the sensibly named "complemented" spectra ($\mathbf{BA}$), these properties get even better.

For nice spectra, like $K(n)$, $BP$, Thom spectra, finite telescopes, etc., have explicit models of these spectra been thought about or are they of any interest? It seems, though I have not yet read Bousfield's original work on it, that $aX$ is a choice of generator for the class of $X$-acyclics. I am curious to know, though it may not be of any practical use, what the homotopy of $aK(n)$ might look like. We know that $K(n)$ is complemented, so we must have $\langle aK(n)\vee K(n)\rangle=\langle S\rangle$, so clearly $\langle aK(n)\rangle\>\langle\bigvee_{m\neq n}K(n)\rangle$.

I guess however that it's actually quite clear what $aT(n)$ (the finite telescope) should be.

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For some reason math stopped showing up after I wrote that last sentence, but $ aT(n)$ should be $\bigvee_{m<n}T(m)\vee F(n+1)$. – Jon Beardsley Oct 21 '12 at 23:03
Did you try putting back-ticks (i.e. ``) around the dollar-signs? – David White Oct 22 '12 at 2:45

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