Real Algebraic and Analytic Geometry |

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.

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