2023/10/20 by Luca Carai, Carai, Luca, Serafina Lapenta +3
Computer Science · Mathematics · #06F20 #26E35 #46A40 #Advanced Algebra and Logic #Advanced Topology and Set Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2310.13427
openalex publication_date 2023/10/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Combining tools from category theory, model theory, and non-standard analysis we extend Baker-Beynon dualities to the classes of all Abelian ℓ-groups and all Riesz spaces (also known as vector lattices). The extended dualities have a strong geometrical flavor, as they involve a non-standard version of the category of polyhedral cones and piecewise (homogeneous) linear maps between them. We further show that our dualities are induced by the functor \mathttSpec, once it is understood how to endow it with "coordinates" in some ultrapower U of ℝ. This also allows us to characterize the topological spaces arising as spectra of Abelian ℓ-groups and Riesz spaces as certain subspaces of Uκ endowed with the Zariski topology given by definable functions in their respective languages. Furthermore, we provide some applications of the extended duality by characterizing, in geometrical terms, semisimplicity, Archimedeanity, and the existence of weak and strong order-units. Finally, we show that our dualities afford a neat and simpler proof of Panti's celebrated characterization of the prime ideals in free Abelian ℓ-groups and Riesz spaces.