Functorial equations for lexicographic products
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- by Franz-Viktor Kuhlmann, Salma Kuhlmann and Saharon Shelah PDF
- Proc. Amer. Math. Soc. 131 (2003), 2969-2976 Request permission
Abstract:
We generalize the main result of an earlier paper by the authors (Exponentiation in power series fields, Proc. Amer. Math. Soc. 125 (1997), 3177–3183) concerning the convex embeddings of a chain $\Gamma$ in a lexicographic power $\Delta ^{\Gamma }$. For a fixed non-empty chain $\Delta$, we derive necessary and sufficient conditions for the existence of non-empty solutions $\Gamma$ to each of the lexicographic functorial equations \[ (\Delta ^{\Gamma })^{\leq 0} \simeq \Gamma \>, \ \ (\Delta ^{\Gamma }) \simeq \Gamma \ \ \mbox { and } \ \ (\Delta ^{\Gamma })^{<0} \simeq \Gamma \;.\]References
- Hausdorff, F.$\,$: Grundzüge der Mengenlehre, Verlag von Veit, Leipzig (1914).
- Holland, W. C. – Kuhlmann, S. – McCleary, S.$\,$: The Arithmetic of Lexicographic Exponentiation, preprint.
- Salma Kuhlmann, Isomorphisms of lexicographic powers of the reals, Proc. Amer. Math. Soc. 123 (1995), no. 9, 2657–2662. MR 1260172, DOI 10.1090/S0002-9939-1995-1260172-4
- Franz-Viktor Kuhlmann, Salma Kuhlmann, and Saharon Shelah, Exponentiation in power series fields, Proc. Amer. Math. Soc. 125 (1997), no. 11, 3177–3183. MR 1402868, DOI 10.1090/S0002-9939-97-03964-6
- Joseph G. Rosenstein, Linear orderings, Pure and Applied Mathematics, vol. 98, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. MR 662564
Additional Information
- Franz-Viktor Kuhlmann
- Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, Saskatchewan, Canada S7N 5E6
- Email: fvk@math.usask.ca
- Salma Kuhlmann
- Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, Saskatchewan, Canada S7N 5E6
- MR Author ID: 293156
- Email: skuhlman@math.usask.ca
- Saharon Shelah
- Affiliation: Department of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel
- MR Author ID: 160185
- ORCID: 0000-0003-0462-3152
- Email: shelah@math.huji.ac.il
- Received by editor(s): June 28, 2001
- Received by editor(s) in revised form: April 16, 2002
- Published electronically: January 2, 2003
- Additional Notes: The first and second authors were partially supported by an NSERC research grant. The third author was partially supported by the Edmund Landau Center for Research in Mathematical Analysis, supported by the Minerva Foundation (Germany); publication number 615
- Communicated by: Carl G. Jockusch, Jr.
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 2969-2976
- MSC (2000): Primary 06A05; Secondary 03C60
- DOI: https://doi.org/10.1090/S0002-9939-03-06830-8
- MathSciNet review: 1993201