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

TL;DR

This paper studies the structure of clones containing addition on the set b{Z}_{pq} for distinct primes p and q, showing the lattice of such clones is finite and providing bounds on its size.

Contribution

It establishes the finiteness of the clone lattice on b{Z}_{pq} containing addition and provides an explicit upper bound for its cardinality.

Findings

The lattice of clones containing addition on b{Z}_{pq} is finite.

An upper bound for the lattice's size is given via injective functions to product lattices.

These clones can be generated by functions of arity at most b{max}(p,q).

Abstract

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 $max(\{p,q\})$.