2025/08/29 by Marco Abbadini, Abbadini, Marco, Daniele Mundici +1
Computer Science · Decision Sciences · Mathematics · #Advanced Algebra and Logic #Fuzzy and Soft Set Theory #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2508.21500
As a topological generalization of the notion of a multiset, a boolean multispace is a boolean space X with a continuous function u\colon X→ \mathbb Z>0, where \mathbb Z>0=\1,2,…\ has the discrete topology. In this paper the category of boolean multispaces and continuous multiplicity-decreasing morphisms with respect to the divisibility order is shown to be dually equivalent to the category of unital Specker ℓ-groups and unital ℓ-homomorphisms. This result extends Stone duality, because unital Specker ℓ-groups whose distinguished unit is singular are equivalent to boolean algebras. Boolean multispaces, in turn, are categorically equivalent to the Priestley duals of the MV-algebras corresponding to unital Specker ℓ-groups via the Γ functor. Via duality, we show that the category of unital Specker ℓ-groups has finite colimits and finite products, but lacks some countable copowers and equalizers.