Integrally closed (Ordered group)
In algebra, a partially ordered group G is called integrally closed if for all elements a and b of G, if an ≤ b for all natural n then a ≤ 1.[1]
This property is somewhat stronger than the fact that a partially ordered group is Archimedean, though for a lattice-ordered group to be integrally closed and to be Archimedean is equivalent.[2] There is a theorem that every integrally closed directed group is already abelian. This has to do with the fact that a directed group is embeddable into a complete lattice-ordered group if and only if it is integrally closed.[1]
References
- ↑ 1.0 1.1 Glass, A. M. W. (22 July 1999). Partially Ordered Groups. ISBN 981449609X. https://books.google.com/books?id=5oTVCgAAQBAJ&pg=PA191.
- ↑ Birkhoff, Garrett (1942). "Lattice-Ordered Groups". The Annals of Mathematics 43 (2): 313. doi:10.2307/1968871. ISSN 0003-486X. http://dx.doi.org/10.2307/1968871.
Further reading
Everett, C. J.; Ulam, S. (1945). "On Ordered Groups". Transactions of the American Mathematical Society 57 (2): 208–216. doi:10.2307/1990202.
Notes
This article incorporates material from partially ordered group on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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