Some more examples of undecidable theories

M. A. Taislin

Algebra i logika, Tome 6 (1967) no. 3, pp. 105-111.

Résumé

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 $A(n,\Lambda )=\{A(n,\tau,\Lambda )| \tau=1,2,\dots\}$, $A^*(n,\Lambda )=\{A^*(n,\tau,\Lambda )| \tau=1,2,\dots\}$ 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.