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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:[email protected]
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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
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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