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This article is cited in 1 scientific paper (total in 1 paper)
Lattices of Interpretability Types of Varieties
D. M. Smirnov
Abstract:
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.
Keywords:
interpretability type, variety, $\Gamma$-divisible Abelian group, module over a normal field, $n$-groupoid.
Received: 14.04.2004
Citation:
D. M. Smirnov, “Lattices of Interpretability Types of Varieties”, Algebra Logika, 44:2 (2005), 198–210; Algebra and Logic, 44:2 (2005), 109–116
Linking options:
https://www.mathnet.ru/eng/al104 https://www.mathnet.ru/eng/al/v44/i2/p198
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