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.