Clone and hypervariety correspondences

Abstract

The general correspondence between clones and hypervarieties was set out by Walter Taylor in [5], By mapping each varietyV to its cloneClone V Taylor obtained inverse isomorphisms between the lattice of all clone varieties and the lattice of all hypervarieties. Taylor also described a more specific correspondence in [6], using the 1-clone Cl₁ V of a varietyV. Since 1-clones are just monoids, he found a one-to-one correspondence from the uncountably infinite lattice of monoid varieties into the lattice of all hypervarieties. In this paper we show how Taylor's construction may be carried out forn-clones, for anyn ≥1. Thus we usen-clones to restrict the general correspondence to a family (α_n) of correspondences, including the monoid correspondence as the special casen=1. These correspondences pick out certain families of hypervarieties, then-closed hypervarieties (forn≥ 1), and we show that there are at least countably many such hypervarieties for eachn. This represents some progress towards the goal of understanding the structure of the lattice of all hypervarieties.