Closed sets of finitary functions between finite fields of coprime order

TL;DR

This paper characterizes certain function sets between finite fields of coprime order, showing they are generated by a single unary function and linked to invariant subspaces under a specific linear transformation.

Contribution

It provides a complete characterization of $(F_p,F_q)$-linearly closed clonoids via invariant subspaces and proves each is generated by one unary function.

Findings

Characterization of function sets via invariant subspaces.

Each function set is generated by a single unary function.

Connection to linear transformations with minimal polynomial $x^{q-1} - 1$.

Abstract

We investigate the finitary functions from a finite field $\mathbb{F}_q$ to the finite field $\mathbb{F}_p$, where $p$ and $q$ are powers of different primes. An $(\mathbb{F}_p,\mathbb{F}_q)$-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 invariant subspaces of the vector space $\mathbb{F}_p^{\mathbb{F}_q\backslash\{0\}}$ with respect to a certain linear transformation with minimal polynomial $x^{q-1} - 1$. Furthermore we prove that each of these subsets of functions is generated by one unary function.