Expansions of abelian squarefree groups

TL;DR

This paper studies the structure and size of certain clones of functions on squarefree cyclic groups, providing bounds and generation results that deepen understanding of their algebraic properties.

Contribution

It establishes the finiteness of the clone lattice containing addition on squarefree groups and provides bounds on its size and generating functions.

Findings

The clone lattice on squarefree groups containing addition is finite.

An upper bound for the lattice size is derived using product of linearly closed clonoids.

Clones can be generated by functions of bounded arity, at most the largest prime factor.

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