Real Algebraic and Analytic Geometry

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94. W. Charles Holland, Salma Kuhlmann, Stephen H.McCleary:
Lexicographic Exponentiation of Chains.

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Submission: 2004, February 19.

Abstract:
The lexicographic power $\Delta ^{\Gamma}$ of chains $\Delta$ and $\Gamma$ is, roughly, the Cartesian power $\prod_{\gamma \in \Gamma}\Delta$, totally ordered lexicographically from the left. Here the focus is on certain powers in which either $\Delta = \R$ or $\Gamma = \R$, with emphasis on when two such powers are isomorphic and on when $\Delta ^{\Gamma}$ is $2$-homogeneous. The main results are:\sn 1) For a countably infinite ordinal $\alpha$, $\R ^{\alpha ^* +\alpha}\simeq\R ^{\alpha}$.\sn 2) $\R ^{\R}\not\simeq\R ^{\Q}$. \sn 3) For $\Delta$ a countable ordinal $\geq 2$, $\Delta ^{\R}$, with its smallest element deleted, is 2-homogeneous.

Mathematics Subject Classification (2000): 06A05, 03C60.

Full text, 23p.: dvi 112k, ps.gz 184k, pdf 282k.


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