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.
Similar content being viewed by others
REFERENCES
R. McKenzie, “On the covering relation in the interpretability lattice of equational theories,” Alg. Univ., 30, No.3, 399–421 (1993).
D. M. Smirnov, “The lattice of interpretability types of Cantor varieties,” Algebra Logika, 43, No.4, 445–458 (2004).
G. Baumslag, “Some aspects of groups with unique roots,” Acta Math., 104, 217–303 (1960).
P. J. Hilton and S. M. Yahya, “Unique divisibility in abelian groups,” Acta Math. Acad. Sc. Hung., 14, 229–239 (1963).
O. C. Garcia and W. Taylor, The Lattice of Interpretability Types of Varieties, Mem. Am. Math. Soc., Vol. 50(305), Am. Math. Soc., Providence, RI (1984).
D. M. Smirnov, “Interpretability types of varieties and strong Mal’tsev conditions,” Sib. Mat. Zh., 35, No.3, 683–695 (1994).
R. McKenzie and S. Swerczkowski, “Non-covering in the interpretability of equational theories,” Alg. Univ., 30, No.2, 157–170 (1993).
R. McKenzie and W. Taylor, “Interpretations of module varieties,” J. Alg., 135, No.2, 456–493 (1990).
B. L. van der Waerden, Algebra. I, Springer, Berlin (1971).
D. M. Smirnov, “Varieties defined by permutations,” Algebra Logika, 39, No.1, 104–118 (2000).
D. M. Smirnov, “An algorithm for constructing a variety of arbitrary finite dimension,” Algebra Logika, 37, No.2, 167–180 (1998).
D. M. Smirnov, “Varieties defined by permutations,” Algebra Logika, 42, No.2, 237–354 (2003).
Additional information
Deceased.
__________
Translated from Algebra i Logika, Vol. 44, No. 2, pp. 198–210, March–April, 2005.
Rights and permissions
About this article
Cite this article
Smirnov, D.M. Lattices of interpretability types of varieties. Algebr Logic 44, 109–116 (2005). https://doi.org/10.1007/s10469-005-0012-1
Received:
Issue Date:
DOI: https://doi.org/10.1007/s10469-005-0012-1