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 Algebra Logika, 2004, Volume 43, Number 4, Pages 445–458 (Mi al82)

The Lattice of Interpretability Types of Cantor Varieties

D. M. Smirnov

Abstract: For integers $1\leqslant m<n$, a Cantor variety with $m$ basic $n$-ary operations $\omega_i$ and $n$ basic $m$-ary operations $\lambda_k$ is a variety of algebras defined by identities $\lambda_k(\omega_1(\bar x),\ldots,\omega_m(\bar x))=x_k$ and $\omega_i(\lambda_1(\bar y),\ldots ,\lambda_n(\bar y))=y_i$, where $\bar x=(x_1,\ldots,x_n)$ and $\bar y=(y_1,\ldots,y_m)$. We prove that interpretability types of Cantor varieties form a distributive lattice, ${\mathbb C}$, which is dual to the direct product ${\mathbb Z}_1\times{\mathbb Z}_2$ of a lattice, ${\mathbb Z}_1$, of positive integers respecting the natural linear ordering and a lattice, ${\mathbb Z}_2$, of positive integers with divisibility. The lattice ${\mathbb C}$ is an upper subsemilattice of the lattice ${\mathbb L}^int$ of all interpretability types of varieties of algebras.

Keywords: Cantor variety, distributive lattice, interpretability types of varieties, lattice of varieties

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English version:
Algebra and Logic, 2004, 43:4, 249–257

Bibliographic databases:

UDC: 512.572

Citation: D. M. Smirnov, “The Lattice of Interpretability Types of Cantor Varieties”, Algebra Logika, 43:4 (2004), 445–458; Algebra and Logic, 43:4 (2004), 249–257

Citation in format AMSBIB
\Bibitem{Smi04} \by D.~M.~Smirnov \paper The Lattice of Interpretability Types of Cantor Varieties \jour Algebra Logika \yr 2004 \vol 43 \issue 4 \pages 445--458 \mathnet{http://mi.mathnet.ru/al82} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2105848} \zmath{https://zbmath.org/?q=an:1115.08008} \transl \jour Algebra and Logic \yr 2004 \vol 43 \issue 4 \pages 249--257 \crossref{https://doi.org/10.1023/B:ALLO.0000035116.89584.14} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-18244394917} 

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This publication is cited in the following articles:
1. D. M. Smirnov, “Lattices of Interpretability Types of Varieties”, Algebra and Logic, 44:2 (2005), 109–116
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