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This article is cited in 13 scientific papers (total in 13 papers)
$\Sigma$-Definability of countable structures over real numbers, complex numbers, and quaternions
A. S. Morozova, M. V. Korovinab a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
b A. P. Ershov Institute of Informatics Systems Sib. Br. RAS
Abstract:
We study $\Sigma$-definability of countable models over hereditarily finite ($\mathbb{HF}$-) superstructures over the field $\mathbb R$ of reals, the field $\mathbb C$ of complex numbers, and over the skew field $\mathbb H$ of quaternions. In particular, it is shown that each at most countable structure of a finite signature, which is $\Sigma$-definable over $\mathbb{HF}(\mathbb R)$ with at most countable equivalence classes and without parameters, has a computable isomorphic copy. Moreover, if we lift the requirement on the cardinalities of the classes in a definition then such a model can have an arbitrary hyperarithmetical complexity, but it will be hyperarithmetical in any case. Also it is proved that any countable structure $\Sigma$-definable over $\mathbb{HF}(\mathbb C)$, possibly with parameters, has a computable isomorphic copy and that being $\Sigma$-definable over $\mathbb{HF}(\mathbb H)$ is equivalent to being $\Sigma$-definable over $\mathbb{HF}(\mathbb R)$.
Keywords:
countable model, computable model, $\Sigma$-definability.
Received: 16.04.2007 Revised: 14.02.2008
Citation:
A. S. Morozov, M. V. Korovina, “$\Sigma$-Definability of countable structures over real numbers, complex numbers, and quaternions”, Algebra Logika, 47:3 (2008), 335–363; Algebra and Logic, 47:3 (2008), 193–209
Linking options:
https://www.mathnet.ru/eng/al362 https://www.mathnet.ru/eng/al/v47/i3/p335
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Abstract page: | 401 | Full-text PDF : | 131 | References: | 39 | First page: | 3 |
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