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 Pi(m)(j=1) F-qj = K to a finite product of finite fields Pi(n)(i=1) F-pi = F, where vertical bar K vertical bar and vertical bar F vertical bar are coprime. An (F, 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 F-p[K-x]-submodules of F-p(K), where K-x is the multiplicative monoid of K = Pi(m)(i=1) F-qi. 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 (F, K)-linearly closed clonoids.