The Lattice of Interpretability Types of Cantor Varieties

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.