We investigate the finitary functions from a finite product of finite fields \(\prod _{j =1}^m\mathbb {F}_{q_j} = {\mathbb K}\) to a finite product of finite fields \(\prod _{i =1}^n\mathbb {F}_{p_i} = {\mathbb {F}}\), 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.

我们研究了从有限域的有限乘积\(\prod _{j =1}^m\mathbb {F}_{q_j} = {\mathbb K}\)到有限域的有限乘积\( \prod _{i =1}^n\mathbb {F}_{p_i} = {\mathbb {F}}\)，其中\(|{\mathbb K}|\)和\(|{\mathbb { F}}|\)互质。一个\（（{\ mathbb {F}}，{\ mathbbķ}）\） -linearly封闭clonoid是这些功能，其根据组合物从右侧和从关闭的子集左侧用线性映射。我们通过\({\mathbb {F}}_p[{\mathbb K}^{\times }]\) - \(\mathbb {F}_p^{{\ mathbb K}}\)，其中\({\mathbb K}^{\times }\)是\({\mathbb K}= \prod _{i=1}^m {\mathbb {F}}_{q_i}\)的乘性幺半群。此外，我们证明这些函数的每个子集都是由一组一元函数生成的，并且我们提供了不同\(({\mathbb {F}},{\mathbb K})\) 的数量的上限 - 线性封闭的克隆体。