Clones containing the Mal'cev operation of ${\mathbb{Z}}_{pq}$

We investigate finitary functions from ${\mathbb{Z}}_{pq}$ to ${\mathbb{Z}}_{pq}$ for two distinct prime numbers $p$ and $q$. We show that the lattice of all clones on the set ${\mathbb{Z}}_{pq}$ which contain the addition of ${\mathbb{Z}}_{pq}$ is finite. We provide an upper bound for the cardinality of this lattice through an injective function to the direct product of the lattice of all $({\mathbb{Z}}_p,{\mathbb{Z}}_q)$-linearly closed clonoids to the $p+1$ power and the lattice of all $({\mathbb{Z}}_q,{\mathbb{Z}}_p)$-linearly closed clonoids to the $q+1$ power. These lattices are studied in arXiv:1910.11759 and there we can find the exact cardinality of them. Furthermore, we prove that these clones can be generated by a set of functions of arity at most $max(\{p,q\})$.