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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| Format: | Article |
| Language: | English |
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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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| _version_ | 1832567641149014016 |
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| author | David J. Foulis |
| author_facet | David J. Foulis |
| author_sort | David J. Foulis |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-317a6fdc2cdf4ca0a85642d07263db4e |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2003-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-317a6fdc2cdf4ca0a85642d07263db4e2025-02-03T01:00:57ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252003-01-012003442787280110.1155/S0161171203210395Archimedean unital groups with finite unit intervalsDavid J. Foulis0Department of Mathematics and Statistics, University of Massachusetts, Amherst 01003, MA, USALet 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.http://dx.doi.org/10.1155/S0161171203210395 |
| spellingShingle | David J. Foulis Archimedean unital groups with finite unit intervals International Journal of Mathematics and Mathematical Sciences |
| title | Archimedean unital groups with finite unit intervals |
| title_full | Archimedean unital groups with finite unit intervals |
| title_fullStr | Archimedean unital groups with finite unit intervals |
| title_full_unstemmed | Archimedean unital groups with finite unit intervals |
| title_short | Archimedean unital groups with finite unit intervals |
| title_sort | archimedean unital groups with finite unit intervals |
| url | http://dx.doi.org/10.1155/S0161171203210395 |
| work_keys_str_mv | AT davidjfoulis archimedeanunitalgroupswithfiniteunitintervals |