On the coprimeness relation from the viewpoint of monadic second-order logic
- Authors: Speranski S.O.1, Pakhomov F.N.1,2
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Affiliations:
- Steklov Mathematical Institute of Russian Academy of Sciences
- Ghent University
- Issue: Vol 86, No 6 (2022)
- Pages: 207-222
- Section: Articles
- URL: https://journal-vniispk.ru/1607-0046/article/view/133918
- DOI: https://doi.org/10.4213/im9340
- ID: 133918
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Abstract
Let $\mathfrak{C}$ denote the structure of the natural numbers with thecoprimeness relation. We prove that for each non-zero natural number $n$,if a $\Pi^1_n$-set of natural numbers is closed under automorphismsof $\mathfrak{C}$, then it is definable in $\mathfrak{C}$ by a monadic$\Pi^1_n$-formula of the signature of $\mathfrak{C}$ having exactly $n$ setquantifiers.On the other hand, we observe that even a much weaker version of this property fails for certain expansions of $\mathfrak{C}$.
About the authors
Stanislav Olegovich Speranski
Steklov Mathematical Institute of Russian Academy of Sciences
Email: katze.tail@gmail.com
Candidate of physico-mathematical sciences, no status
Fedor Nikolaevich Pakhomov
Steklov Mathematical Institute of Russian Academy of Sciences; Ghent University
Email: pakhfn@gmail.com
Candidate of physico-mathematical sciences, Senior Researcher
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