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:ℤn→G and ρ:G→ℤκ linking G with the simplicial groups ℤn and ℤκ.We show that ξ is...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2003-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171203210395 |
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| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |