Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-27T09:58:39.514Z Has data issue: false hasContentIssue false
  • English
  • Français

The Hahn representation theorem for ℓ-groups in ZFA

Published online by Cambridge University Press:  12 March 2014

D. Gluschankof
D. Gluschankof*
Affiliation:
U.R.A. 753, Équipe De Logique-Mathématique, U.F.R. De Mathématiques, Université Paris7, 2, Pl. Jussieu, 75251 Paris Cedex 05, France, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

In [7] the author discussed the relative force —in the set theory ZF— of some representation theorems for ℓ-groups (lattice-ordered groups). One of the theorems not discussed in that paper is the Hahn representation theorem for abelian ℓ-groups. This result, originally proved by Hahn (see [8]) for totally ordered groups and half a century later by Conrad, Harvey and Holland for the general case (see [4]), states that any abelian ℓ-group can be embedded in a Hahn product of copies of R (the real line with its natural totally-ordered group structure). Both proofs rely heavily on Zorn's Lemma which is equivalent to AC (the axiom of choice).

The aim of this work is to point out the use of non-constructible axioms (i.e., AC and weaker forms of it) in the proofs. Working in the frame of ZFA, that is, the Zermelo-Fraenkel set theory where a non-empty set of atoms is allowed, we present alternative proofs which, in the totally ordered case, do not require the use of AC. For basic concepts and notation on ℓ-groups the reader can refer to [1] and [2]. For set theory, to [11].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 2 , June 2000 , pp. 519 - 524
Copyright
Copyright © Association for Symbolic Logic 2000

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]Anderson, M. and Feil, T., Lattice-Ordered Groups, An Introduction, D. Reidel Publishing Company, Dordrecht, 1988.CrossRefGoogle Scholar
[2]Bigard, A., Keimel, K., and Wolfenstein, S., Groupes et Anneaux Réticulés, LNM608, Springer-Verlag, Berlin, 1977.CrossRefGoogle Scholar
[3]Burris, S. and Sankappanavar, H. P., A Course in Universal Algebra, Springer-Verlag, New York - Heidelberg - Berlin, 1981.CrossRefGoogle Scholar
[4]Conrad, P., Harvey, J., and Holland, C., The Hahn embedding theorem for abelian lattice-ordered groups, Transactions of the American Mathematical Society, (1963), no. 108, pp. 143–169.Google Scholar
[5]Felgner, U. and Jech, T., Variants of the axiom of choice in set theory with atoms, Fundamenta Mathematicae, (1973), no. 79, pp. 79–85.Google Scholar
[6]Gluschankof, D., Prime ideal and Sikorski extension theorems for some ℓ-groups, Ordered Algebraic Structures (Martinez, J., editor), Kluwer Academic Publishers, 1989, pp. 113–122.CrossRefGoogle Scholar
[7]Gluschankof, D., On the relative force of the representation theorems for ℓ-groups, Algebra Universalis, (1995), no. 34, pp. 380–390.Google Scholar
[8]Hahn, H., Über die nichtarchimedischen Gröβensysteme, Sitz. ber. K. Akad. der Wiss., Math. Nat. Kl., vol. IIa (1907), no. 116, pp. 601–655.Google Scholar
[9]Halpern, J. D., The independence of the Axiom of Choice from the Boolean prime ideal theorem, Fundamenta Mathematicae, (1964), no. 55, pp. 57–66.Google Scholar
[10]Halpern, J. D., On a question of Tarski and a maximal theorem of Kurepa, Pacific Journal of Mathematics, vol. 41 (1972), pp. 111–121.CrossRefGoogle Scholar
[11]Jech, T. J., The Axiom of Choice, North-Holland, Amsterdam-London, 1973.Google Scholar