The Lattice of Interpretability Types of Cantor Varieties

Abstract

For integers 1 ≤ m < n, a Cantor variety with m basic n-ary operations ωi and n basic m-ary operations λk is a variety of algebras defined by identities λk(ω1(\(x\)), ... , ωm(\(\bar x\))) = \(x\)k and ωi(λ1(\(\bar y\)), ... ,λn(\(\bar y\))) = yi, where \(\bar x\) = (x1., ... , xn) and \(\bar y\) = (y1, ... , ym). 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.