Geodesic
Lattices of Interpretability Types of Varieties
Algebra i logika, Tome 44 (2005) no. 2, pp. 198-210.

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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, module over a normal field
Mots-clés : $\Gamma$-divisible Abelian group, $n$-groupoid. 
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     author = {D. M. Smirnov},
     title = {Lattices of {Interpretability} {Types} of {Varieties}},
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D. M. Smirnov. Lattices of Interpretability Types of Varieties. Algebra i logika, Tome 44 (2005) no. 2, pp. 198-210. http://geodesic.mathdoc.fr/item/AL_2005_44_2_a3/

[1] R. McKenzie, “On the covering relation in the interpretability lattice of equational theories”, Algebra Univers., 30:3 (1993), 399–421 | DOI | MR | Zbl

[2] D. M. Smirnov, “Reshetka tipov interpretiruemosti mnogoobrazii Kantora”, Algebra i logika, 43:4 (2004), 445–458 | MR | Zbl

[3] G. Baumslag, “Some aspects of groups with unique roots”, Acta Math., 104 (1960), 217–303 | DOI | MR | Zbl

[4] P. J. Hilton, S. M. Yahya, “Unique divisibility in abelian groups”, Acta Math. Acad. Sci. Hung., 14 (1963), 229–239 | DOI | MR | Zbl

[5] O. C. Garcia, W. Taylor, The lattice of interpretability types of varieties, Mem. Am. Math. Soc., 50 (305), Am. Math. Soc., Providence, RI, 1984 | MR

[6] D. M. Smirnov, “Tipy interpretiruemosti mnogoobrazii i strogie usloviya Maltseva”, Sib. matem. zh., 35:3 (1994), 683–695 | MR | Zbl

[7] R. McKenzie, S. Swierczkowski, “Non-covering in the interpretability lattice of equational theories”, Algebra Univers., 30:2 (1993), 157–170 | DOI | MR | Zbl

[8] R. McKenzie, W. Taylor, “Interpretations of module varieties”, J. Algebra, 135:2 (1990), 456–493 | DOI | MR | Zbl

[9] B. L. van der Varden, Algebra, Nauka, M., 1976 | MR

[10] D. M. Smirnov, “Mnogoobraziya, opredelimye podstanovkami”, Algebra i logika, 39:1 (2000), 104–118 | MR | Zbl

[11] D. M. Smirnov, “Algoritm postroeniya mnogoobraziya proizvolno zadannoi konechnoi razmernosti”, Algebra i logika, 37:2 (1998), 167–180 | MR | Zbl

[12] D. M. Smirnov, “O mnogoobraziyakh, opredelimykh podstanovkami”, Algebra i logika, 42:2 (2003), 237–354 | MR