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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Describing groups
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by Meng-Che Ho PDF
Proc. Amer. Math. Soc. 145 (2017), 2223-2239 Request permission

Abstract:

We study two complexity notions of groups – the syntactic complexity of a computable Scott sentence and the $m$-degree of the index set of a group. Finding the exact complexity of one of them usually involves finding the complexity of the other, but this is not always the case. Knight et al. determined the complexity of index sets of various structures.

In this paper, we focus on finding the complexity of computable Scott sentences and index sets of various groups. We give computable Scott sentences for various different groups, including nilpotent groups, polycyclic groups, certain solvable groups, and certain subgroups of $\mathbb {Q}$. In some of these cases, we also show that the sentences we give are optimal. In the last section, we also show that d-$\Sigma _2\subsetneq \Delta _3$ in the complexity hierarchy of pseudo-Scott sentences, contrasting the result saying d-$\boldsymbol {\Sigma }_2=\boldsymbol {\Delta }_3$ in the complexity hierarchy of Scott sentences, which is related to the boldface Borel hierarchy.

References
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Additional Information
  • Meng-Che Ho
  • Affiliation: Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI 53706
  • MR Author ID: 1200055
  • ORCID: setImmediate$0.7583630476368097$9
  • Email: turboho@gmail.com
  • Received by editor(s): April 6, 2016
  • Received by editor(s) in revised form: June 2, 2016
  • Published electronically: January 31, 2017
  • Communicated by: Mirna Džamonja
  • © Copyright 2017 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 2223-2239
  • MSC (2010): Primary 03D45, 03C57, 20F10
  • DOI: https://doi.org/10.1090/proc/13458
  • MathSciNet review: 3611333