Archimedean unital groups with finite unit intervals

David J Foulis

doi:10.1155/s0161171203210395

Abstract

Let G be a unital group with a finite unit interval E, let n be the number of atoms in E, and let κ be the number of extreme points of the state space Ω(G). We introduce canonical order‐preserving group homomorphisms ξ : ℤn → G and ρ : G → ℤκ linking G with the simplicial groups ℤn and ℤκ.We show that ξ is a surjection and ρ is an injection if and only if G is torsion‐free. We give an explicit construction of the universal group (unigroup) for E using the canonical surjection ξ. If G is torsion‐free, then the canonical injection ρ is used to show that G is Archimedean if and only if its positive cone is determined by a finite number of homogeneous linear inequalities with integer coefficients.

Highlights

Introduction and basic definitionsIn this paper, we continue the study of unital groups with finite unit intervals begun in [2, 3]
We continue the study of unital groups with finite unit intervals begun in [2, 3]
If G is a unital group with unit interval E, and if K is an abelian group, a mapping φ : E → K is called a K-valued measure on E if and only if p, q, p + q ∈ E ⇒ φ(p + q) = φ(p) + φ(q)

Summary

ARCHIMEDEAN UNITAL GROUPS WITH FINITE UNIT INTERVALS

Let G be a unital group with a finite unit interval E, let n be the number of atoms in E, and let κ be the number of extreme points of the state space Ω(G). If G is torsion-free, the canonical injection ρ is used to show that G is Archimedean if and only if its positive cone is determined by a finite number of homogeneous linear inequalities with integer coefficients. If G is a partially ordered abelian group and u ∈ G+, we define the interval E := G+[0, u] := {g ∈ G | 0 ≤ g ≤ u}. If G is a unital group with unit interval E, and if K is an abelian group, a mapping φ : E → K is called a K-valued measure on E if and only if p, q, p + q ∈ E ⇒ φ(p + q) = φ(p) + φ(q). If every K-valued measure on E is the restriction to E of a group homomorphism from G to K, G is called a K-unital group.

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