Closed sets of finitary functions between products of finite fields of coprime order

Abstract

We investigate the finitary functions from a finite product of finite fields ${\prod }_{j=1}^m{\mathbb{F}}_{q_j}=\mathbb{K}$ to a finite product of finite fields ${\prod }_{i=1}^n{\mathbb{F}}_{p_i}=\mathbb{F}$, where $|\mathbb{K}|$ and $|\mathbb{F}|$ are coprime. An $(\mathbb{F},\mathbb{K})$-linearly closed clonoid is a subset of these functions which is closed under composition from the right and from the left with linear mappings. We give a characterization of these subsets of functions through the ${\mathbb{F}}_p[{\mathbb{K}}^{\times }]$-submodules of ${\mathbb{F}}_p^{\mathbb{K}}$, where ${\mathbb{K}}^{\times }$ is the multiplicative monoid of $\mathbb{K}={\prod }_{i=1}^m{\mathbb{F}}_{q_i}$. Furthermore we prove that each of these subsets of functions is generated by a set of unary functions and we provide an upper bound for the number of distinct $(\mathbb{F},\mathbb{K})$-linearly closed clonoids.