Hostname: page-component-745bb68f8f-s22k5 Total loading time: 0 Render date: 2025-02-10T05:19:35.051Z Has data issue: false hasContentIssue false
  • English
  • Français

CONTRIBUTIONS TO THE THEORY OF F-AUTOMATIC SETS

Part of: Model theory Sequences and sets Theory of computing

Published online by Cambridge University Press:  08 June 2021

CHRISTOPHER HAWTHORNE
CHRISTOPHER HAWTHORNE*
Affiliation:
DEPARTMENT OF PURE MATHEMATICS UNIVERSITY OF WATERLOO 200 UNIVERSITY AVENUE WEST WATERLOO, ONN2L 3G1, CANADAE-mail:cdchawthorne@uwaterloo.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

Fix an abelian group $\Gamma $ and an injective endomorphism $F\colon \Gamma \to \Gamma $ . Improving on the results of [2], new characterizations are here obtained for the existence of spanning sets, F-automaticity, and F-sparsity. The model theoretic status of these sets is also investigated, culminating with a combinatorial description of the F-sparse sets that are stable in $(\Gamma ,+)$ , and a proof that the expansion of $(\Gamma ,+)$ by any F-sparse set is NIP. These methods are also used to show for prime $p\ge 7$ that the expansion of $(\mathbb {F}_p[t],+)$ by multiplication restricted to $t^{\mathbb {N}}$ is NIP.

Keywords

automatic setsparsestableNIPF-set

MSC classification

Primary: 03C45: Classification theory, stability and related concepts
Secondary: 11B85: Automata sequences 68Q45: Formal languages and automata
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 1 , March 2022 , pp. 127 - 158
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bell, J., Ghioca, D., and Moosa, R., Effective isotrivial Mordells–Lang in positive characteristic, preprint, 2020, arXiv:2010.08579.Google Scholar
Bell, J. and Moosa, R., F-sets and finite automata . Journal de Théorie des Nombres de Bordeaux , vol. 31 (2019), no. 1, pp. 101–130.CrossRefGoogle Scholar
Bruyere, V., Hansel, G., Michaux, C., and Villemaire, R., Logic and p-recognizable sets of integers . Bulletin of the Belgian Mathematical Society Simon Stevin , vol. 1 (1994), pp. 191–238.Google Scholar
Conant, G. and Laskowski, M. C., Weakly minimal groups with a new predicate . Journal of Mathematical Logic , vol. 20 (2020), no. 2, p. 2050011.10.1142/S0219061320500117CrossRefGoogle Scholar
Hawthorne, C. D. C., Automata and tame expansions of (Z,+), preprint, 2020, arXiv:2007.00070.Google Scholar
Moosa, R. and Scanlon, T., F-structures and integral points on semiabelian varieties over finite fields . American Journal of Mathematics , vol. 126 (2004), no. 3, pp. 473–522.CrossRefGoogle Scholar
Yu, S., Regular languages , Handbook of Formal Languages: Volume 1 Word, Language, Grammar (G. Rozenberg and A. Salomaa, editors), Springer, Berlin and Heidelberg, 1997, pp. 41–110.CrossRefGoogle Scholar