The Lattice of Interpretability Types of Cantor Varieties

Abstract

For integers 1 ≤ m < n, a Cantor variety with m basic n-ary operations ω_i and n basic m-ary operations λ_k is a variety of algebras defined by identities λ_k(ω₁($x$), ... , ω_m($\bar x$)) = $x$ _k and ωi(λ₁($\bar y$), ... ,λ_n($\bar y$)) = y _i, where $\bar x$ = (x ₁., ... , x _n) and $\bar y$ = (y ₁, ... , y _m). We prove that interpretability types of Cantor varieties form a distributive lattice, ℂ, which is dual to the direct product ℤ₁ × ℤ₂ of a lattice, ℤ₁, of positive integers respecting the natural linear ordering and a lattice, ℤ₂, of positive integers with divisibility. The lattice ℂ is an upper subsemilattice of the lattice $\mathbb{L}^{\operatorname{int} }$ of all interpretability types of varieties of algebras.