Lattices of interpretability types of varieties
| Autor: | D. M. Smirnov |
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| Rok vydání: | 2005 |
| Předmět: | |
| Zdroj: | Algebra and Logic. 44:109-116 |
| ISSN: | 1573-8302 0002-5232 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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