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 Algebra i Logika. Sem., 1967, Volume 6, Number 3, Pages 105–111 (Mi al1112)  Some more examples of undecidable theories

M. A. Taislin

Abstract: 1. Let $L(\sigma)$ be a class of all relational systems of finite type $\sigma$. Suppose $\sigma'$ be a type which includes the type $\sigma$ and $D_{\sigma'}\ne D_{\sigma}$. Let $\sigma'(\Lambda )=1$ whenever $\Lambda \in D_{\sigma'}\setminus D_{\sigma}$. Let $K\subset L(\sigma)$ and $K(\sigma')=\{M\in L(\sigma')| M\upharpoonright \sigma\in K\}$. It is for a number of classes $K\subset L(\sigma)$ that the elementary theory of class $K(\sigma')$ is hereditarily undecidable. This holds for example, if class $K\subset L(\sigma)$ satisfies the conditions 1.-3.
2. When denoting $A(n,\tau,\Lambda )$ resp. $A^*(n,\tau,\Lambda )$ free algebras with $n$ free generators in the class of associative commutative $\tau$-nilpotent algebras over field $\Lambda$ resp. in the class of associative $\tau$-nilpotent algebras over field $\Lambda$ and putting $A(n,\Lambda )=\{A(n,\tau,\Lambda )| \tau=1,2,…\}$, $A^*(n,\Lambda )=\{A^*(n,\tau,\Lambda )| \tau=1,2,…\}$ it is proved that the elementary theories of the classes $A(n,\Lambda )$, $A^*(n,\Lambda )$ are hereditarily undecidable for $n\geqslant2$ if $\Lambda$ is field of characteristic $0$ and for $n\geqslant 3$ in each other cases. In all cases the elementary theory of class $A^*(2,\Lambda )$ is hereditarily undecidable. Full text: PDF file (249 kB)

Bibliographic databases: Citation: M. A. Taislin, “Some more examples of undecidable theories”, Algebra i Logika. Sem., 6:3 (1967), 105–111 Citation in format AMSBIB
\Bibitem{Tai67} \by M.~A.~Taislin \paper Some more examples of undecidable theories \jour Algebra i Logika. Sem. \yr 1967 \vol 6 \issue 3 \pages 105--111 \mathnet{http://mi.mathnet.ru/al1112} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=0224470} 

• http://mi.mathnet.ru/eng/al1112
• http://mi.mathnet.ru/eng/al/v6/i3/p105

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This publication is cited in the following articles:
1. A. I. Kokorin, A. G. Pinus, “Decidability problems of extended theories”, Russian Math. Surveys, 33:2 (1978), 53–96    •   Contact us: math-net2021_12 [at] mi-ras ru Terms of Use Registration to the website Logotypes © Steklov Mathematical Institute RAS, 2021