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 $\max(p_1,\dots,p_m)$.