Expansions of abelian squarefree groups

Abstract

We investigate finitary functions from ${\mathbb{Z}}_n$ to ${\mathbb{Z}}_n$ for a squarefree number $n$. We show that the lattice of all clones on the squarefree set ${\mathbb{Z}}_{p_1\cdots p_m}$ which contain the addition of ${\mathbb{Z}}_{p_1\cdots p_m}$ is finite. We provide an upper bound for the cardinality of this lattice through an injective function to the direct product of the lattices of all $({\mathbb{Z}}_{p_i},{\mathbb{F}}_i)$-linearly closed clonoids, $\mathcal{L}({\mathbb{Z}}_{p_i},{\mathbb{F}}_i)$, to the $p_i+1$ power, where ${\mathbb{F}}_i={\prod }_{j\in \{1,\dots ,m\}\backslash \{i\}}{\mathbb{Z}}_{p_j}$. These lattices are studied in the litterature and we can find an upper bound for cardinality of them. Furthermore, we prove that these clones can be generated by a set of functions of arity at most $\max (p_1,\dots ,p_m)$.

arXiv admin note: substantial text overlap with arXiv:2006.00291