Lattices of Interpretability Types of Varieties

D. M. Smirnov

Algebra i logika, Tome 44 (2005) no. 2, pp. 198-210

Résumé

Let $\Pi$ be the set of all primes, $\mathbb A$ the field of all algebraic numbers, and $Z$ the set of square-free natural numbers. We consider partially ordered sets of interpretability types such as $$ \mathbb L_\Pi=(\{[AD_\Gamma]\mid\Gamma\subseteq\Pi\},\le), \qquad \mathbb L_\mathbb A=(\{[M_\mathbb K]\mid\mathbb K\subseteq\mathbb A\},\le), $$ and $$ \mathbb L_Z=(\{[G_n]\mid n\in Z\},\le), $$ where $AD_\Gamma$ is a variety of $\Gamma$-divisible Abelian groups with unique taking of the $p$th root $\xi_p(x)$ for every $p\in\Gamma$, $M_\mathbb K$ is a variety of $\mathbb K$-modules over a normal field $\mathbb K$, contained in $\mathbb A$, and $G_n$ is a variety of $n$-groupoids defined by a cyclic permutation $(12\ldots n)$. We prove that $\mathbb L_\Pi$, $\mathbb L_\mathbb A$, and $\mathbb L_Z$ are distributive lattices, with $\mathbb L_\Pi\cong \mathbb L_\mathbb A\cong \mathbb S\rm ub\,\Pi$ and $\mathbb L_Z\cong \mathbb S\rm ub_f\Pi$ where $\mathbb S\rm ub\,\Pi$ and $\mathbb S\rm ub_f\Pi$ are lattices (w. r. t. inclusion) of all subsets of the set $\Pi$ and of finite subsets of $\Pi$, respectively.

Détail

Keywords: interpretability type, variety, module over a normal field
Mots-clés : $\Gamma$-divisible Abelian group, $n$-groupoid.