Unital Specker $ll$-groups and boolean multispaces

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.

💡 Research Summary

The paper introduces a topological generalisation of multisets called Boolean multispace: a Boolean space X (compact, totally disconnected Hausdorff) equipped with a continuous function u : X → ℤ_{>0} (positive integers with the discrete topology). The value u(x) is interpreted as the multiplicity of the point x, and because the codomain is discrete, each u has a finite image.

A category Bms is defined whose objects are pairs (X,u) and whose morphisms γ : X → Y are continuous maps satisfying the multiplicity‑decreasing condition: for every x∈X the integer u_Y(γ(x)) divides u_X(x). The associated “ratio” ζ_γ = u_X · u_Y(γ)^{‑1} is itself a continuous ℤ_{>0}‑valued function.

On the algebraic side, the authors work with ℓ‑groups (lattice‑ordered abelian groups) equipped with a distinguished unit u ≥ 0, i.e. an element that bounds every group element after multiplication by a suitable positive integer. A Specker ℓ‑group is an ℓ‑group generated (as a group) by its singular elements—those s ≥ 0 such that for all 0 ≤ a ≤ s we have a ∧ (s − a) = 0. In a Specker ℓ‑group all non‑zero singular elements coincide, and the group is order‑isomorphic to ℤ once the natural order on ℤ is imposed.

Key structural facts are proved:

• For any maximal ideal m of a Specker ℓ‑group S there exists a unique surjective ℓ‑homomorphism ρ_m : S → ℤ whose kernel is m (Theorem 2.4). Moreover ρ_m sends the greatest singular element s_S to 1 (Lemma 2.5).

• For each g∈S the map g^♮ : µ(S) → ℤ defined by g^♮(m)=ρ_m(g) is continuous and takes only finitely many values (Theorem 2.8). Consequently µ(S) (the space of maximal ideals of S) is a Boolean space, and the assignment g↦g^♮ yields an isomorphism of unital ℓ‑groups (S,u) ≅ (C(µ(S)), u^♮).

Thus the classical Stone duality (Boolean algebras ↔ Boolean spaces) is recovered when the distinguished unit of a Specker ℓ‑group is singular; the present result extends it to arbitrary units.

The central construction is the functor S : Bms → uSℓg, where uSℓg denotes the category of unital Specker ℓ‑groups and unit‑preserving ℓ‑homomorphisms. On objects S sends (X,u) to the unital ℓ‑group (C(X), u), where C(X) consists of all continuous ℤ‑valued functions on X and the unit is the given multiplicity function. On morphisms γ : (X,u_X) → (Y,u_Y) it acts by \