Unital Specker $\ell$-groups and boolean multispaces
Marco Abbadini, Daniele Mundici
TL;DR
The paper constructs a duality between boolean multispaces $\mathsf{Bms}$ and the opposite of unital Specker $\ell$-groups $uS\ell g^{op}$ via the functor $\mathcal{S}$, extending Stone duality through a topological multispace perspective. It also shows that the functor $\mathcal{B}$ completes the duality, yielding a full adjoint dual equivalence between $\mathsf{Bms}$ and $uS\ell g$, with connections to Specker MV-algebras via the $\Gamma$ functor. The finite-limit structure of $\mathsf{Bms}$ contrasts with its lack of certain infinite limits, and dually $uS\ell g$ and $SMV$ miss some countable copowers and equalizers, highlighting precise limits/colimits behavior. Closure properties within $ulg$ reveal that infinite coproducts need not remain Specker, indicating structural boundaries of Specker objects under standard categorical constructions. Overall, the work ties topological multisets to algebraic Specker frameworks, offering a unified view through dualities and clarifying the role of MV-algebras and Priestley duality in this setting.
Abstract
As a topological generalization of the notion of a multiset, a boolean multispace is a boolean space $X$ with a continuous function $u\colon X\to \mathbb Z_{>0}$, where $\mathbb Z_{>0}=\{1,2,\dots\}$ 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 $\ell$-groups and unital $\ell$-homomorphisms. This result extends Stone duality, because unital Specker $\ell$-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 $\ell$-groups via the $Γ$ functor. Via duality, we show that the category of unital Specker $\ell$-groups has finite colimits and finite products, but lacks some countable copowers and equalizers.