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

Abstract

We investigate the finitary functions from a finite product of finite fields prod _{j =1}^mmathbb {F}_{q_j} = {mathbb K} to a finite product of finite fields prod _{i =1}^nmathbb {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.