Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T08:28:46.954Z Has data issue: false hasContentIssue false
  • English
  • Français

An embedding theorem for ordered groups

Published online by Cambridge University Press:  17 April 2009

Colin D. Fox
Colin D. Fox
Affiliation:
Department of Mathematics, La Trobe University, Bundoora, Victoria.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that if the normal, closure of an element a, of an orderable group, G, is abelian, then G can be embedded in an orderable group, G#, which contains an n-th root of a for every positive integer, n. Furthermore, every order of G extends to an order of G#.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 12 , Issue 3 , June 1975 , pp. 321 - 335
Copyright
Copyright © Australian Mathematical Society 1975

References

[1]Блудов, B.B., Медведев, H.Я. [Bludov, B.B., Medvedev, N.Ja.], “О пополнении упорядочиваемых метабелевых групп” [On the completion of an orderable metabelian group], Algebra i Logika 13 (1974), 369–373.Google ScholarPubMed
[2]Chehata, C.G., “On a theorem on ordered groups”, Proc. Glasgow Math. Assoc. 4 (1958), 16–21.CrossRefGoogle Scholar
[3]Conrad, Paul, “Extensions of ordered groups”, Proc. Amer. Math. Soc. 6 (1955), 516–528.CrossRefGoogle Scholar
[4]Fox, Colin D., “The problem of adjoining roots to ordered groups”, (PhD thesis, Australian National University, Canberra, 1974). See also Abstract: Bull. Austral. Math. Soc. 11 (1974), 157–158.Google Scholar
[5]Fuchs, L., Partially ordered algebraic systems (Pergamon, Oxford, London, New York, Paris, 1963).Google Scholar
[6]Fuchs, László, Teilweise geordnete algebraische Strukturen (Akadémiai Kiadó, Budapest, 1966).Google Scholar
[7]Каргаполов, М.И., Мерзляков, Ю.И., Ремесленников, В.Н., [Kargapolov, M.I., Merzljakov, Yu.I., Remeslennikov, V.N.], Коуровская тетрадь (Нерешенные задачи теори групп) [Kourov notebook. Unsolved problems in the theory of groups, 3rd edition, supplemented] (Izdat. Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk, 1969).Google Scholar
[8]Кокорин, А.И., Копытов, В.М., Линейно упорядоченные групы (Nauka, Moscov, 1972). Kokorin, A.I. and Kopytov, V.M., Fully ordered groups (translated by Louvish, D.. John Wiley & Sons, New York, Toronto; Israel Program for Scientific translations, Jerusalem, London, 1974).Google Scholar
[10]Копытов, В.М. [Kopytov, V.M.], “О пополнении центра упорядоченной группы” [On the completion of the centre of an ordered group], Ural. Gos. Univ. Mat. Zap. 4 (1963), 20–24.Google ScholarPubMed
[11]Мальцев, А.И. [A. I. Mal'cev] “Нильпотентные группы без кручения” [Nilpotent torsion-free groups], Izv. Akad. Nauk SSSR Ser. Mat. 13 (1949), 201–212.Google ScholarPubMed
[12]Мальцев, А.И. [Mal'cev, A. I.] “О доупорядочени групп” [On the completion of group order], Trudy Mat. Inst. Steklov. 38 (1951), 173–175.Google ScholarPubMed
[13]Minassian, Donald P., “An embedding theorem for ordered groups”, Canad. J. Math. 24 (1972), 944–946.CrossRefGoogle Scholar
[14]Смирнов, Д.М. [Smirnov, D.M.], “Правоупорядоченные группы” [Right-ordered groups], Algebra i Logika 5 (6) (1966), 41–59.Google ScholarPubMed