Archimedean unital groups with finite unit intervals

LetGbe a unital group with a finite unit intervalE, letnbe the number of atoms inE, and letκbe the number of extreme points of the state spaceΩ(G). We introduce canonical order-preserving group homomorphismsξ:ℤn→Gandρ:G→ℤκlinkingGwith the simplicial groupsℤnandℤκ.We show thatξis a surjection andρis an injection if and only ifGis torsion-free. We give an explicit construction of the universal group (unigroup) forEusing the canonical surjectionξ. IfGis torsion-free, then the canonical injectionρis used to show thatGis Archimedean if and only if its positive cone is determined by a finite number of homogeneous linear inequalities with integer coefficients.