Closed sets of finitary functions between finite fields of coprime order

Abstract

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

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

“…Then the set of functions that are polymorphisms of all pairs of all the relations in R is denoted by Pol(R). As in [7,Lemma 5.5] we can use polymorphisms to prove the following. Lemma 4.5.…”

“…In the [7] there is a complete description of the structure of all (F, K)-linearly closed clonoids in case F and K are fields and the results we will present are a generalization of this description.…”

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

“…In [6] there is a complete description of the structure of all (F, K)-linearly closed clonoids in case F and K are fields and the results we will present are a generalization of this description.…”