Finite sublattices in the lattice of clones

Abstract

The lattice$\mathcal{L}_k$ of clones of functions over a k-element set is studied. It is shown that every lattice which is a countable direct product of finite lattices is embedded (up to isomorphism) in$\mathcal{L}_4$ and, hence, in$\mathcal{L}_k$ for k ≥ 4. This directly implies that every finite and any countable residually finite lattice is embedded in$\mathcal{L}_k$, k ≥ 4, and that no nontrivial quasi-identity holds in$\mathcal{L}_k$, k ≥ 4. A number of particular lattices (which are free in some lattice varieties) embeddable in$\mathcal{L}_k$, k ≥ 4,are presented.