International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 44, Pages 2787-2801
doi:10.1155/S0161171203210395
Archimedean unital groups with finite unit intervals
Department of Mathematics and Statistics, University of Massachusetts, Amherst 01003, MA, USA
Received 23 October 2002
Copyright © 2003 David J. Foulis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.