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 Sib. Èlektron. Mat. Izv., 2020, Volume 17, Pages 1013–1026 (Mi semr1270)

Mathematical logic, algebra and number theory

A note on decidable categoricity and index sets

N. Bazhenov, M. Marchuk

Sobolev Institute of Mathematics, 4, Acad. Koptyug Ave., Novosibirsk, 630090, Russia

Abstract: A structure $S$ is decidably categorical if $S$ has a decidable copy, and for any decidable copies $A$ and $B$ of $S$, there is a computable isomorphism from $A$ onto $B$. Goncharov and Marchuk proved that the index set of decidably categorical graphs is $\Sigma^0_{\omega+2}$ complete. In this paper, we isolate two familiar classes of structures $K$ such that the index set for decidably categorical members of $K$ has a relatively low complexity in the arithmetical hierarchy. We prove that the index set of decidably categorical real closed fields is $\Sigma^0_3$ complete. We obtain a complete characterization of decidably categorical equivalence structures. We prove that decidably presentable equivalence structures have a $\Sigma^0_4$ complete index set. A similar result is obtained for decidably categorical equivalence structures.

Keywords: decidable categoricity, autostability relative to strong constructivizations, index set, real closed field, equivalence structure, strong constructivization, decidable structure.

 Funding Agency Grant Number Ministry of Science and Higher Education of the Russian Federation 075-15-2019-1613 The work is supported by Mathematical Center in Akademgorodok under agreement No. 075-15-2019-1613 with the Ministry of Science and Higher Education of the Russian Federation.

DOI: https://doi.org/10.33048/semi.2020.17.076

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Bibliographic databases:

UDC: 510.5
MSC: 03D45
Received April 28, 2020, published July 28, 2020
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Citation: N. Bazhenov, M. Marchuk, “A note on decidable categoricity and index sets”, Sib. Èlektron. Mat. Izv., 17 (2020), 1013–1026

Citation in format AMSBIB
\Bibitem{BazMar20} \by N.~Bazhenov, M.~Marchuk \paper A note on decidable categoricity and index sets \jour Sib. \Elektron. Mat. Izv. \yr 2020 \vol 17 \pages 1013--1026 \mathnet{http://mi.mathnet.ru/semr1270} \crossref{https://doi.org/10.33048/semi.2020.17.076} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000557456400001} `