Algebra Universalis, 33 (1995) 458-465 0002-5240/95/040458-08501.50 + 0.20/0 9 1995 Birkh/iuser Verlag, Basel Clone and hypervariety correspondences S. L. WISMATH* Abstract. The general correspondence between clones and hypervarieties was set out by Walter Taylor in [5]. By mapping each variety ~,/to its clone CloneTU, 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 Cll(V) of a variety ~. 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 for n-clones, for any n > 1. Thus we use n-clones to restrict the general correspondence to a family (aN) of correspondences, including the monoid correspondence as the special case n = 1. These correspondences pick out certain families of hypervarieties, the n-closed hypervarieties (for n >- I), and we show that there are at least countably many such hypervarieties for each n. This represents some progress towards the goal of understanding the structure of the lattice of all hypervarieties. 1. Introduction The general correspondence between clones and hypervarieties was set out by W. Taylor in [5]. By mapping each variety ~ to its clone Clone~t/~ 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 Cll (~) of a variety V: since l-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 construc- tion may be carried out for n-clones, for any n > 1. Thus we restrict the general correspondence to a family (en :n -> 1) of correspondences, including the monoid correspondence as the special case n -- 1. These correspondences pick out certain families of hypervarieties, called the n-closed hypervarieties, and we show that there are at least countably many such hypervarieties for each n. This represents some progress towards the goal of understanding the structure of the lattice of all hypervarieties. Presented by W. Taylor. Received March 22, 1993; accepted in final form January 17, 1994. * Research supported by NSERC of Canada. 458