The structure of the Bousfield lattice
by Mark Hovey and John Palmieri
Using Ohkawa's theorem that the collection of Bousfield classes is a
set, we perform a number of constructions with Bousfield classes. In
particular, we describe a greatest lower bound operator; we also note
that a certain subset DL of the Bousfield lattice is a frame, and we
examine some consequences of this observation. We make several
conjectures about the structure of the Bousfield lattice and DL. In
particular, we conjecture that DL is obtained by killing "strange"
spectra, such as the Brown-Comenetz dual of the sphere. We introduce a
new "Boolean algebra of spectra" cBA, which contains Bousfield's BA and
is complete. Our conjectures allow us to identify cBA as being
isomorphic to the complete atomic Boolean algebra on {K(n) : n>= 0},
{A(n) : n>= 2}, and HF_p. Our conjectures imply that BA is the
subBoolean algebra consisting of finite wedges of the K(n) and A(n), and
their complements.