Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-27T23:20:55.603Z Has data issue: false hasContentIssue false
  • English
  • Français

Construction of orders in Abelian groups

Published online by Cambridge University Press:  24 October 2008

H.-H. Teh
H.-H. Teh
Affiliation:
Queen's UniversityBelfast
Get access Rights & Permissions [Opens in a new window]

Extract

Let G be an Abelian group. A binary relation ≥ denned in G is called an order of G if for each x, y, z ε G,

(i) x ≥ y or y ≥ x (and hence x ≥ x);

(ii) x ≥ y and y ≥ x ⇒ x = y,

(if x ≥ y and x ≠ y, we write x > y);

(iii) x ≥ y and y ≥ z ⇒ x = z;

(iv) z ≥ y ⇒ x + z ≥ y + z.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 57 , Issue 3 , July 1961 , pp. 476 - 482
Copyright
Copyright © Cambridge Philosophical Society 1961

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Baer, R.The subgroup of elements of finite order of an abelian group. Ann. Math. 37 (1936), 766–81.CrossRefGoogle Scholar
(2)Birkhoff, G.Lattice theory. Revised ed. (New York, 1960).Google Scholar
(3)Clifford, A. H.Note on Hahn's Theorem on ordered abelian groups. Proc. Amer. Math. Soc. 5 (1954), 860–3.Google Scholar
(4)Everett, C. J.Note on a result of L. Fuchs's on ordered groups. Amer. J. Math. 72 (1950), 216.CrossRefGoogle Scholar
(5)Fuchs, L.On the extension of the partial order of groups. Amer. J. Math. 72, (1950), 191–4.CrossRefGoogle Scholar
(6)Gravett, K. A. H.Valued linear spaces. Quart. J. Math. (2), 6 (1955), 309–14.CrossRefGoogle Scholar
(7)Gravett, K. A. H.Ordered abelian groups. Quart. J. Math. (2), 7 (1956), 55–63.CrossRefGoogle Scholar
(8)Hahn, H.Über die nichtarchimedischen Grossensysteme. S.B. Akad Wiss. Wien, 116, (1907), 601–55.Google Scholar
(9)Kurosh, A. G.The theory of groups (New York, 1956).Google Scholar