Closed sets of finitary functions between products of finite fields of pair-wise coprime order
Stefano Fioravanti · arXiv (Cornell University) · 2020
We investigate the finitary functions from a finite product of finite fields $\prod_{j =1}^m\mathbb{F}_{q_j} = K$ to a finite product of finite fields $\prod_{i =1}^n\mathbb{F}_{p_i} = F$, where $p_1,\dots,p_n$, $q_1,\dots,q_m$ are powers of different primes. 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 invariant subspaces of the vector spaces $\mathbb{F}_{p_i}^{\prod_{j =1}^m\mathbb{F}_{q_j}}$ with respect to a certain set of linear transformations. 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 clonoid.