Closed sets of finitary functions between finite fields of coprime order

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.

arXiv comment16 pages