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Archimedean unital groups with finite unit intervals. (English) Zbl 1033.06008
Author’s abstract: “Let \(G\) be a unital group with a finite unit interval \(E\), let \(n\) be the number of atoms in \(E\), and let \(\kappa\) be the number of extreme points of the stage space \(\Omega(G)\). We introduce canonical order-preserving group homomorphisms \(\xi : \mathbb Z^n\rightarrow G\) and \(\rho : G\rightarrow \mathbb Z^\kappa\) linking \(G\) with the simplicial groups \(\mathbb Z^n\) and \(\mathbb Z^\kappa\). We show that \(\xi\) is a surjection and \(\rho\) 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 \(\xi\). If \(G\) is torsion-free, then the canonical injection \(\rho\) 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.”
Several connections between this theory and the theory of MV-algebras (or the theory of effect algebras, respectively) are described.

MSC:
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
06D35 MV-algebras
03G12 Quantum logic
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