Generalized Bousfield Lattices and a Generalized Retract Conjecture

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.