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

We investigate the finitary functions from a finite product of finite fields to a finite product of finite fields , where and are coprime. An -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 -submodules of , where is the multiplicative monoid of . 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 -linearly closed clonoids.