• Corpus ID: 219176640

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

@article{Fioravanti2020ClonesCT,
  title={Clones containing the Mal'cev operation of \$\mathbb\{Z\}\_\{pq\}\$},
  author={Stefano Fioravanti},
  journal={arXiv: Rings and Algebras},
  year={2020}
}

• S. Fioravanti
• Published 30 May 2020
• Mathematics
• arXiv: Rings and Algebras

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