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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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