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Chebyshevskii Sb., 2020, Volume 21, Issue 1, Pages 165–178 (Mi cheb866)  

On elementary theories of algebraically closed groups

V. G. Durnev, O. V. Zetkina, A. I. Zetkina

P.G. Demidov Yaroslavl State University

Abstract: In paper for any algebraically closed group $G$, as well as for the class of the algebraically closed groups, we prove algorithmic undecidability of the positive $\forall^2 \exists^{24}$-theory and $\forall^3 \exists^{2}$-theory. For an arbitrary $g\in G$, we also prove the decidability of the equation of the type
$$ w(x_1, \ldots , x_n) = g, $$
where $w(x_1, \ldots , x_n)$ is a non-empty irreducible word in the unknowns $x_1,\ldots x_n\in G$.

Keywords: algebraically closed group, positive theory, equation.

DOI: https://doi.org/10.22405/2226-8383-2018-21-1-165-178

Full text: PDF file (623 kB)
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UDC: 512+512.5+512.54+512.54.03

Citation: V. G. Durnev, O. V. Zetkina, A. I. Zetkina, “On elementary theories of algebraically closed groups”, Chebyshevskii Sb., 21:1 (2020), 165–178

Citation in format AMSBIB
\Bibitem{DurZetZet20}
\by V.~G.~Durnev, O.~V.~Zetkina, A.~I.~Zetkina
\paper On elementary theories of algebraically closed groups
\jour Chebyshevskii Sb.
\yr 2020
\vol 21
\issue 1
\pages 165--178
\mathnet{http://mi.mathnet.ru/cheb866}
\crossref{https://doi.org/10.22405/2226-8383-2018-21-1-165-178}


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