Hostname: page-component-7bb8b95d7b-l4ctd Total loading time: 0 Render date: 2024-09-12T07:07:04.615Z Has data issue: false hasContentIssue false
  • English
  • Français

Lexicographic exponentiation of chains

Published online by Cambridge University Press:  12 March 2014

W. C. Holland ,
S. Kuhlmann  and
S. H. McCleary
W. C. Holland
Affiliation:
Department of Mathematics and Statistics, Bowling Green State University, Bowling Green. Ohio 43403., USA, E-mail: [email protected] Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OHIO 43403., USA
S. Kuhlmann
Affiliation:
Research Unit Algebra and Logic, University of Saskatchewan, Mclean Hall, 106 Wiggins Road, Saskatoon. SK S7N 5E6., Canada, E-mail: [email protected] Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OHIO 43403., USA
S. H. McCleary
Affiliation:
Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OHIO 43403., USA 7180 Las Vistas Las Cruces, New Mexico 88005., USA, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The lexicographic power ΔΓ of chains Δ and Γ is, roughly, the Cartesian power ΠγЄΓΔ totally ordered lexicographically from the left. Here the focus is on certain powers in which either Δ = ℝ or ℚ = ℝ, with emphasis on when two such powers are isomorphic and on when ΔΓ is 2-homogeneous. The main results are:

(1) For a countably infinite ordinal

(2) ℝℝ ≄ ℝℚ.

(3) For Δ a countable ordinal ≥ 2, Δℝ with its smallest element deleted, is 2-homogeneous.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 2 , June 2005 , pp. 389 - 409
Copyright
Copyright © Association for Symbolic Logic 2005

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] Fuchs, L., Partially ordered algebraic systems, Addison-Wesley, Reading, Mass., 1963.Google Scholar
[2] Green, T., Properties of chain products and Ehrenfeucht-Fraïssé games on chains, Msc. Thesis, University of Saskatchewan, 08 2002.Google Scholar
[3] Hausdoff, F. Grundzüge einer Theorie der geordneten Mengen, Mathematische Annalen, vol. 65 (1908), pp. 435–505.CrossRefGoogle Scholar
[4] Hausdoff, F., Grundzüge der Mengenlehre, Verlag von Veit, Leipzig, 1914.Google Scholar
[5] Kuhlmann, F.-V., Kuhlmann, S., and Shelah, S., Exponentiation in power series fields, Proceedings of the American Mathematical Society, vol. 125 (1997), no. 11, pp. 3177–3183.CrossRefGoogle Scholar
[6] Kuhlmann, F.-V., Kuhlmann, S., and Shelah, S., Functorial equations for lexicographic products, Proceedings of the American Mathematical Society, vol. 131 (2003), pp. 2969–2976.CrossRefGoogle Scholar
[7] Kuhlmann, S., Isomorphisms of lexicographic powers of the reals, Proceedings of the American Mathematical Society, vol. 123 (1995). no. 9, pp. 2657–2662.CrossRefGoogle Scholar
[8] Kuhlmann, S., Ordered exponential fields, The Fields Institute Monograph Series, vol. 12, 2000.Google Scholar
[9] Kuhlmann, S. and Shklah, S., κ-hounded exponential-logarithmic power series fields, 2004, submitted.Google Scholar
[10] Rosknstein, J. G., Linear orderings, Academic Press, New York-London, 1982.Google Scholar
[11] Warton, P., Lexicographic powers of the real line, Ph. D. Dissertation, Bowling Green State University, 1998.Google Scholar