AbstractWe investigate the finitary functions from a finite product of finite fields $$\prod _{j =1}^m\mathbb {F}_{q_j} = {\mathbb K}$$
∏
j
=
1
m
F
q
j
=
K
to a finite product of finite fields $$\prod _{i =1}^n\mathbb {F}_{p_i} = {\mathbb {F}}$$
∏
i
=
1
n
F
p
i
=
F
, where $$|{\mathbb K}|$$
|
K
|
and $$|{\mathbb {F}}|$$
|
F
|
are coprime. An $$({\mathbb {F}},{\mathbb K})$$
(
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 $${\mathbb {F}}_p[{\mathbb K}^{\times }]$$
F
p
[
K
×
]
-submodules of $$\mathbb {F}_p^{{\mathbb K}}$$
F
p
K
, where $${\mathbb K}^{\times }$$
K
×
is the multiplicative monoid of $${\mathbb K}= \prod _{i=1}^m {\mathbb {F}}_{q_i}$$
K
=
∏
i
=
1
m
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 $$({\mathbb {F}},{\mathbb K})$$
(
F
,
K
)
-linearly closed clonoids.