Generalized Bousfield lattices and a generalized retract conjecture (Q473127)

The authors record the definition of a \textit{Bousfield lattice} [\textit{A. K. Bousfield}, Comment. Math. Helv. 54, 368--377 (1979; Zbl 0421.55002)] and they indicate the definition of a \textit{retract conjecture}, cf. [\textit{M. Hovey} and \textit{J. F. Palmieri}, in: Meyer, Jean-Pierre (ed.) et al., Homotopy invariant algebraic structures. A conference in honor of J. Michael Boardman. AMS special session on homotopy theory, Baltimore, MD, USA, January 7-10, 1998. Providence, RI: American Mathematical Society. Contemp. Math. 239, 175--196 (1999; Zbl 0947.55014)]. Their main objective is the generalization of both items from special categories to monoids. They have achieved the following theorem: The generalized retract conjecture holds on the stable homotopy category of harmonic spectra. Details are too involved and overloaded by notations to be recorded here.