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This article is cited in 1 scientific paper (total in 1 paper)
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 ${\mathbb L}^{\rm int}$ of all interpretability types of varieties of algebras.
Keywords:
Cantor variety, distributive lattice, interpretability types of varieties, lattice of varieties.
Received: 12.03.2003
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
Linking options:
https://www.mathnet.ru/eng/al82 https://www.mathnet.ru/eng/al/v43/i4/p445
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