4 Rich groups, weak second-order logic, and applications

Kharlampovich, Olga; Myasnikov, Alexei; Sohrabi, Mahmood

doi:10.1515/9783110719710-004

Cited by 6 publications

(3 citation statements)

References 30 publications

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“…Nies ( [49], [48], etc. ), N. Avni, A. Lubotzky, C. Meiri ( [4], [5], also see [32]), A. Myasnikov, O. Kharlampovich and M. Sohrabi ( [34], [45], [46], etc. ), D. Segal and K. Tent ( [55]), B. Kunyavskii, E. Plotkin, N. Vavilov ( [39]) and others appeared (see [52] for a survey of results).…”

Section: Introduction Historical Overviewmentioning

confidence: 99%

“…A. Khelif ( [36], see also [34]) realized that one can use biinterpretability of a finitely generated structure A with (Z, +×) as a general method to prove that A is QFA. Somewhat later, T. Scanlon (see [53]) independently used this method to show that each finitely generated field is QFA.…”

Section: Introduction Historical Overviewmentioning

confidence: 99%

“…To extend this Zilber's result for the case of rings it is crucial to introduce a notion of regular bi-interpretability. If A is interpreted in B uniformly with respect to a ∅-definable subset D ⊆ B k then we say that A is regularly interpretable in B and write in this case A ∼ = Γ(B, ϕ), provided D is defined by ϕ in B (see [34]). Definition 1.4.…”

Section: Introduction Historical Overviewmentioning

confidence: 99%

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Regular bi-interpretability of Chevalley groups over local rings

Bunina

2022

Preprint

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In this paper we prove that if $G(R)=G_\pi (\Phi,R)$ $(E(R)=E_{\pi}(\Phi, R))$ is an (elementary) Chevalley group of rank $> 1$, $R$ is a local ring (with $\frac{1}{2}$ for the root systems ${\mathbf A}_2, {\mathbf B}_l, {\mathbf C}_l, {\mathbf F}_4, {\mathbf G}_2$ and with $\frac{1}{3}$ for ${\mathbf G}_{2})$, then the group $G(R)$ (or $(E(R)$) is regularly bi-interpretable with the ring~$R$.As a consequence of this theorem, we show that the class of all Chevalley groups over local rings (with the listed restrictions) is elementary definable, i.\,e., if for an arbitrary group~$H$ we have $H\equiv G_\pi(\Phi, R)$, than there exists a ring $R'\equiv R$ such that $H\cong G_\pi(\Phi,R')$.

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