Pointfree spectra of ℓℓ-modules

The three well-known spectra often associated to an ordered ring are: Brumfiel, Keimel, and the maximal spectrum. The pointfree versions of these spectra have been studied for ff-rings [B. Banaschewski, Pointfree topology and the spectra of ff-rings, in: Ordered Algebraic Structures (Curacoa, 1995), Kluwer Acad. Publ., Dordrecht, 1997, pp. 123–148], and the last two spectra for Riesz spaces [M.M. Ebrahimi, A. Karimi, M. Mahmoudi, Pointfree spectra of Riesz space, Appl. Categ. Structures 12 (2004) 397–409]. In this paper we consider an ff-module MM on an ordered ring AA and study the pointfree version of the last two spectra together with the frame CL(M)CL(M) of closed ℓℓ-ideals. We show, among other things, that the pointfree maximal spectrum SL(M)SL(M) and the frame CL(M)CL(M) are completly regular and that, under some conditions, these two spectra are naturally isomorphic, and hence functorial.