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}_qbackslash {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.