Archimedean unital groups with finite unit intervals

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.