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Lattices of interpretability types of varieties

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Abstract

Let Π 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 \(\mathbb{L}_\Pi = (\{ [AD_\Gamma ]|\Gamma \subseteq \Pi \} , \leqslant ),\mathbb{L}_\mathbb{A} = (\{ [M_\mathbb{K} ]|\mathbb{K} \subseteq \mathbb{A}\} , \leqslant )\), and \(\mathbb{L}_Z = (\{ [G_n ]|n \in Z\} , \leqslant )\), where ADΓ is a variety of Γ-divisible Abelian groups with unique taking of the pth root ξp(x) for every p ∈ Γ, \(M_\mathbb{K}\) is a variety of \(\mathbb{K}\)-modules over a normal field \(\mathbb{K}\), contained in \(\mathbb{A}\), and Gn is a variety of n-groupoids defined by a cyclic permutation (12 ...n). We prove that \(\mathbb{L}_\Pi ,\mathbb{L}_\mathbb{A}\), and \(\mathbb{L}_Z\) are distributive lattices, with \(\mathbb{L}_\Pi \cong \mathbb{L}_\mathbb{A} \cong \mathbb{S}ub\;\Pi\) and \(\mathbb{L}_Z \cong \mathbb{S}ub_f \Pi\) where \(\mathbb{S}\)ub Π and \(\mathbb{S}\)ubfΠ are lattices (w.r.t. inclusion) of all subsets of the set Π and of finite subsets of Π, respectively.

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Deceased.

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Translated from Algebra i Logika, Vol. 44, No. 2, pp. 198–210, March–April, 2005.

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Smirnov, D.M. Lattices of interpretability types of varieties. Algebr Logic 44, 109–116 (2005). https://doi.org/10.1007/s10469-005-0012-1

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  • DOI: https://doi.org/10.1007/s10469-005-0012-1

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