Expansions of abelian square-free groups

We investigate finitary functions from n to n for a square-free number n. We show that the lattice of all clones on the square-free set pm which contain the addition of p1⋯pm 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 (pi-linearly closed clonoids, (pi, i), to the pi + 1 power, where i =j {1, m}{i}pj. These lattices are studied in [S. Fioravanti, Closed sets of finitary functions between products of finite fields of pair-wise coprime order, preprint (2020), arXiv:2009.02237] and there we can find an upper bound for their cardinality. Furthermore, we prove that these clones can be generated by a set of functions of arity at most max(p1pm). ;copy: 2021 World Scientific Publishing Company.