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Functorial equations for lexicographic products

Author(s): Franz-Viktor Kuhlmann; Salma Kuhlmann; Saharon Shelah
Journal: Proc. Amer. Math. Soc. 131 (2003), 2969-2976.
MSC (2000): Primary 06A05; Secondary 03C60
Posted: January 2, 2003
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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

\begin{displaymath}(\Delta ^{\Gamma})^{\leq 0} \simeq \Gamma\>, (\Delta ^{\Gam... ...Gamma \mbox{ and } (\Delta ^{\Gamma})^{<0} \simeq \Gamma\;.\end{displaymath}

References:

[H]
Hausdorff, F.$\,$: Grundzüge der Mengenlehre, Verlag von Veit, Leipzig (1914).

[H-K-M]
Holland, W. C. - Kuhlmann, S. - McCleary, S.$\,$: The Arithmetic of Lexicographic Exponentiation, preprint.

[K]
Kuhlmann, S.$\,$: Isomorphisms of Lexicographic Powers of the Reals, Proc. Amer. Math. Soc. 123 (1995), 2657-2662. MR 95k:06001

[K-K-S]
Kuhlmann, F.-V. - Kuhlmann, S. - Shelah, S.$\,$: Exponentiation in power series fields, Proc. Amer. Math. Soc. 125 (1997), 3177-3183. MR 97m:12007

[R]
Rosenstein, J. G.$\,$: Linear orderings, Academic Press, New York - London (1982). MR 84m:06001

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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
Email: skuhlman@math.usask.ca

Saharon Shelah
Affiliation: Department of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel
Email: shelah@math.huji.ac.il

DOI: 10.1090/S0002-9939-03-06830-8
PII: S 0002-9939(03)06830-8
Received by editor(s): June 28, 2001
Received by editor(s) in revised form: April 16, 2002
Posted: 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 of article: Copyright 2003, American Mathematical Society

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