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

TL;DR

This paper characterizes and analyzes the structure of finitary functions between products of finite fields with coprime sizes, focusing on their closure properties and generating sets.

Contribution

It provides a characterization of linearly closed clonoids via module theory and proves they are generated by unary functions, with bounds on their quantity.

Findings

Characterization of function subsets through module theory.

Proof that these subsets are generated by unary functions.

Establishment of an upper bound for the number of such subsets.

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.