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 ) . We introduce canonical order-preserving group homomorphisms ξ : ℤ n → G and ρ : G → ℤ κ linking G with the simplicial groups ℤ n and ℤ κ .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.