An Analogue of the Dichotomy Conjecture on Monoidally Distributive Posets

Abstract

Hovey [in: The Čech centennial (Boston, MA, 1993), 1995, 225–250] proposed the dichotomy conjecture on the stable homotopy category of spectra. Hovey and Palmieri [in: Homotopy invariant algebraic structures (Baltimore, MD, 1998), 1999, 175–196] proved many interesting facts around the dichotomy conjecture from the viewpoint of the Bousfield lattice. The author, Shimomura and Tatehara [Publ. Res. Inst. Math. Sci. 50 (2014), 497–513] defined the notion of monoidally distributive posets as a generalization of the Bousfield lattice. In this paper we consider an analogue of the dichotomy conjecture on monoidally distributive posets, and prove several results around the analogue.