From Freudenthal’s spectral theorem to projectable hulls of unital Archimedean lattice-groups, through compactifications of minimal spectra

Abstract

We use a landmark result in the theory of Riesz spaces -Freudenthal's 1936 Spectral Theorem -to canonically represent any Archimedean lattice-ordered group G with a strong unit as a (non-separating) lattice-group of real valued continuous functions on an appropriate G-indexed zero-dimensional compactification w G Z G of its space Z G of minimal prime ideals. The two further ingredients needed to establish this representation are the Yosida representation of G on its space X G of maximal ideals, and the well-known continuous surjection of Z G onto X G . We then establish our main result by showing that the inclusion-minimal extension of this representation of G that separates the points of Z G -namely, the sublattice subgroup of C (Z G ) generated by the image of G along with all characteristic functions of clopen (closed and open) subsets of Z G which are determined by elements of G -is precisely the classical projectable hull of G. Our main result thus reveals a fundamental relationship between projectable hulls and minimal spectra, and provides the most direct and explicit construction of projectable hulls to date. Our techniques do require the presence of a strong unit.

Orders on groups, and spectral spaces of lattice-groups

“…In connection with Theorem 7.1 we ought to at least mention Speed's paper [21] for distributive lattices, and Conrad's and Martinez' paper [7] for ℓ-groups. Let us also mention that, in the Archimedean case, compactifications of minimal spectra of lattice-groups were recently shown to be inextricably related to the construction of projectable hulls, see [1,14]. We endow 2 G with the smallest topology containing all sets X(a) and X c (a).…”

“…This, as far as we know, is folklore; see [11,Theorem 6.8] for a proof. 1 The result indicates that orders on groups play a rôle in topological dynamics. For more on this, besides Section 6.5 of Ghys' beautiful survey [11], see the research monograph [4].…”

The Chinese Remainder Theorem for Strongly Semisimple MV-Algebras and Lattice-Groups

“…Indeed, spectra of MV-algebras always are completely normal-which affords the existence of the map λ used in the proof above-whereas spectra of rings are not, in general. For more on the important rôle that the continuous retraction λ in (12) plays in the theory of lattice-groups and MV-algebras, see [2] and the references therein.…”