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In [1], Bousfield studied a lattice (Bousfield lattice) on the stable homotopy category of spectra, and in [5], Hovey and Palmieri made the retract conjecture on the lattice. In this paper we generalize the Bousfield lattice and the retract conjecture to the ones on a monoid. We also determine the structure of typical examples of them, which satisfy the generalized retract conjecture. In particular we give the structure of the Bousfield lattice of the stable homotopy category of harmonic spectra explicitly.

For each p-local spectrum E, LE denotes the full subcategory consisting of Elocal spectra of the category of p-local spectra. The Picard group Pic(LE )i s the collection of isomorphism classes of invertible spectra in LE . If this is as et, it is a group with multiplication defined by the smash product. We show that if a spectrum

Let E*'*(X) denote the l?2-term of the Adams-Novikov spectral sequence converging to the homotopy groups of X. We can define the Greek letter elements in E*'*(LnS°) in the same way as those in E*'*(S°). We show that these of E*'*(LnS°) are permanent cycles, though we know little for E%'*(S°).