Closed sets of finitary functions between finite fields of coprime order

We investigate the finitary functions from a finite field $\mathbb{F}_q$ to the finite field $\mathbb{F}_p$, where $p$ and $q$ are powers of different primes. An $(\mathbb{F}_p,\mathbb{F}_q)$-linear 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 space $\mathbb{F}_p^{q-1}$ with respect to a certain linear transformation with minimal polynomial $x^{q-1} - 1$. Furthermore we prove that each of these subsets of functions is generated by one unary function.


Introduction
The problem of characterizing sets of functions that satisfy some closure properties plays an increasingly important role in General Algebra. E. Post's characterization of all clones on a two-element set [Pos41] is a foundational result in this field, which was developed further, e. g., in [Ros69,PK79,Sze86,Leh10]. Starting from [BJK05], clones are used to study the complexity of certain constrain satisfaction problems (CSPs).
The aim of this paper is to describe sets of functions from F q to F p that are closed with linear mappings from the left and from the right, in the case p and q are powers of distinct primes. We are dealing with sets of functions with different domains and codomains; such sets are investigated, e. g., in [AM16] and are called clonoids. Let B be an algebra, and let A be a non-empty set. For a subset C of n∈N B A n and k ∈ N, we let C [k] := C ∩ B A k . According to Definition 4.1 of [AM16] we call C a clonoid with source set A and target algebra B if (1) for all k ∈ N: C [k] is a subuniverse of B A k , and (2) for all k, n ∈ N, for all (i 1 , . . . , i k ) ∈ {1, . . . , n} k , and for all c ∈ C [k] , the function c ′ : A n → B with c ′ (a 1 , . . . , a n ) := c(a i 1 , . . . , a i k ) lies in C [n] . In particular we are interested in those clonoids whose target algebra is the vector space F p which are closed under composition with linear mappings also from the right side.
Definition 1.1. Let p and q be powers of different primes, and let F p and F q be two fields of orders p and q. A (F p , F q )-linear closed clonoid is a non-empty subset C of n∈N F F n q p with the following properties: (1) for all n ∈ N, f, g ∈ C [n] and a, b ∈ F p : af + bg ∈ C [n] ; (2) for all m, n ∈ N, f ∈ C [m] and A ∈ F m×n q : g : (x 1 , . . . , x n ) → f (A · (x 1 , . . . , x n ) t ) is in C [n] .
Clonoids are of interest since they naturally arise in the study of promise constraint satisfaction problems (PCSPs). These problems are investigated, e. g., in [BG18], and recently in [BKO18] clonoid theory has been used to give an algebraic approach to PCSPs. Moreover, a description of the set of all (F p , F q )linear closed clonoids is a useful tool to investigate (polynomial) clones on Z p ×Z q or to represent polynomial functions of semidirect products of groups. In [Kre19] S. Kreinecker characterized linearly closed clonoids on F p , where p is a prime, and found a description of all clones on F p that contain the addition, all iterative algebras on Z p which are closed under composition with the clone generated by + from both sides, and proved that there are infinitely many non-finitely generated clones above Clo(F p × F p , +) for p > 2.
Our main result (Theorem 1.3) provides a complete description of the structure of the lattice of all (F p , F q )-linear closed clonoids. First, an important observation is that each such clonoid is generated by its subset of unary members (Theorem 6.2). Nevertheless, we can also say more about the generators of an (F p , F q )-linear closed clonoid.
Theorem 1.2. Every (F p , F q )-linear closed clonoid is generated by one unary function.
The proof of this result is given in Section 7. With Theorem 6.2 and the characterization of the invariant subspaces lattice of a cyclic linear transformation over a finite-dimensional vector space in [BF67], we obtain a description of the lattice of all (F p , F q )-linear closed clonoids as a direct product of chains (Section 7).
The structure of the lattice of all (F p , F q )-linear closed clonoids depends on the prime factorization of the polynomial g = x q−1 − 1 in F p [x]. Once known this factorization, it is easy to find this lattice. Let us denote by 2 the two-element chain and, in general, by C k the chain with k elements. Moreover, we denote by L(p, q) the lattice of all (F p , F q )-linear closed clonoids. Theorem 1.3. Let p and q be powers of different primes. Let n i=1 p k i i be the factorization of the polynomial g = x q−1 − 1 in F p [x] into its irreducible divisors. Then the number of distinct (F p , F q )-linear closed clonoids is 2 n i=1 (k i + 1) and the lattice of all (F p , F q )-linear closed clonoids, L(p, q), is isomorphic to 2 × n i=1 C k i +1 .

Preliminaries and notations
We use boldface letters for vectors, e. g., u = (u 1 , . . . , u n ) for some n ∈ N. Moreover, we will use v , u for the scalar product of the vectors v and u.
We write C (p,q) (S) for the (F p , F q )-linear closed clonoid generated by a set of functions S. The (F p , F q )-linear closed clonoids form a lattice with the intersection as meet and the (F p , F q )-linear closed clonoid generated by the union as join. The top element of the lattice is the (F p , F q )-linear closed clonoid of all functions and the bottom element consists of only the constant zero functions. Let f be an n-ary function from a group G 1 to a group G 2 . We say that As examples of non-trivial (F p , F q )-linear closed clonoids we can see that the set of all 0-preserving finitary functions from F q to F p forms an (F p , F q )-linear closed clonoid and the set of all finitary functions from F n q to F p that are constant on every line of their domain which passes through the origin, but may have different values at zero, form an (F p , F q )-linear closed clonoid.
Definition 2.1. Let p and q be powers of primes and let f be a function from F n q to F p . The function f is a star function if and only if for every vector w ∈ F n q there exists k ∈ F p such that for every λ ∈ F q \{0}: It is easy to see that the star functions form an (F p , F q )-linear closed clonoid for every p and q and they represent an instance of the nice behaviour that the (F p , F q )-linear closed clonoids have in relation to the lines of the space F n q . Indeed, the composition with linear mappings from the right hand side can be used to permute the values that functions of (F p , F q )-linear closed clonoid have in lines that pass through the origin.

Preliminaries from linear algebra
In this section we review some of concepts of linear algebra that we need in order to find a description of the lattice of all (F p , F q )-linear closed clonoids. We recall that a T-invariant subspace of a linear operator T of a vector space V is a subspace W of V that is preserved by T ; that is, T (W ) ⊆ W . Let S be a set of linear operators of a vector space V. We can consider the S-invariant subspaces lattice of V and we denote it by L(S).
We will sometimes denote a linear operator T by its associated matrix. In Section 7 we will see that the problem to find the structure of the lattice of all (F p , F q )-linear closed clonoids can be reduced to the problem to find all Tinvariant subspaces of the vector space F q−1 p , where T is certain linear transformation that permutes the components of F q−1 p . In [BF67] the structure of the invariant subspaces lattice of a linear transformation on a finite-dimensional vector space over an arbitrary field has been studied, and in [Fri11] the number of invariant subspaces of a finite vector space with respect to a linear operator is determined.
Let T be a linear transformation on a finite-dimensional vector space V over a field K and let g be the minimal polynomial of T . We call T primary if g = p c for some irreducible polynomial p and some positive integer c. We know from [BF67, Theorem 1] that, with the prime factorization of g = s i=1 p k i i over K[x ], we can split the vector space V into what is called its primary decomposition: are called the primary components of V. According to [BF67], the lattice L(T ) of the T -invariant subspaces of V is a direct product of the lattices L(T i ), where T i = T | V i . Thus: In [BF67, Lemma 2] it is proved that L(T ) is a chain if and only if T is cyclic and primary. In particular they show that if the minimal polynomial of T is g = p n , with p irreducible, then:

Interpolation
In this section we want to introduce the concept of interpolation in an (F p , F q )linear closed clonoid. As well as in the Lagrange interpolation formula we have the Lagrange basis polynomials, in the case of (F p , F q )-linear closed clonoids we will have the Lagrange interpolation functions, which are functions built to have a value different from zero only in one point, and they can be seen as characteristic functions of a point in the vector space F n q with codomain {0, 1} ⊆ F p . Definition 4.1. Let a = (a 1 , . . . , a n ) ∈ F n q . The n-ary Lagrange interpolation function f a from F q to F p is the function defined by: be the line of the space F n q generated by the vector b ∈ L. Let us consider l j ∈ F n q for 1 ≤ j ≤ n − 1 such that the solutions of the system formed by the equations ( l j , y = 0) 1≤j≤n−1 describe the line L of F n q . Thus and the claim holds.
The functions above provide the basis to build a generalization of Lagrange interpolation. We show that we can interpolate all 0-preserving functions with the functions of Lemma 4.2.
Lemma 4.3. Let n ∈ N, a = (1, 0, . . . , 0) ∈ F n q , and let C be an (F p , F q )linear closed clonoid which contains f a . Then C contains all n-ary 0-preserving functions from F q to F p .
Proof. Let C = C (p,q) ({f a }) and let f be an n-ary function from F q to F p . Then from Lemma 4.2 we have that We can represent f as: Hence f ∈ C.

A set of generators
In this section we produce a unary 0-preserving function which generates every other 0-preserving function in an (F p , F q )-linear closed clonoid. This is possible choosing the Lagrange interpolation function f 1 .
Theorem 5.1. Let p and q be two powers of distinct primes. Then the (F p , F q )linear closed clonoid of all 0-preserving functions is generated by the unary Lagrange interpolation function f 1 (Definition 4.1).
Proof. Let f be an n-ary 0-preserving function from F q to F p . Let us prove by induction on the arity n that f ∈ C (p,q) ({f 1 }).
Case n = 1: by Lemma 4.3, every 0-preserving unary function is generated by f 1 .
Case n > 1: suppose that the claim holds for n − 1. The strategy is to use the (n−1)-ary 0-preserving functions in C (p,q) ({f 1 }) to interpolate the n-ary function f (1,0,..,0) (Definition 4.1). Then Lemma 4.3 implies that C contains every n-ary 0-preserving function from F q to F p .
6. The generators of every (F p , F q )-linear closed clonoid In this section our aim is to find an analogue of Theorem 5.1 for a generic (F p , F q )-linear closed clonoid C, which allows us to generate C with a set of unary functions. In general we will see that it is the unary part of an (F p , F q )linear closed clonoid that determines the clonoid. To this end we shall show the following Lemma. 1, 0, . . . , 0)) for all λ ∈ F q and f (x ) = g(y ) = 0 for all x ∈ F n q \{λb | λ ∈ F q } and y ∈ F n q \{λ(1, 0, . . . , 0) | λ ∈ F q }. Then f ∈ C (p,q) ({g}).
Proof. Let n ∈ N and b = (b 1 , . . . , b n ) ∈ F n q \{(0, . . . , 0)}. Let 1 ≤ i ≤ n be such that b i = 0 and let f, g : F n q → F p be functions as in the hypothesis. Moreover, let L = {sb | s ∈ F q } be the line of the space F n q generated by the vector b. Let us consider l j ∈ F n q for 1 ≤ j ≤ n − 1 such that the solutions of the system formed by the equations ( l j , y = 0) 1≤j≤n−1 describe the line L of F n q . Then: Hence g ∈ C (p,q) ({g}) and the claim holds.
We are now ready to prove that an (F p , F q )-linear closed clonoid C is generated by its unary part.
Theorem 6.2. Let p and q be powers of different primes. Then every (F p , F q )linear closed clonoid C is generated by its unary functions. Thus C = C (p,q) (C [1] ).
Proof. The inclusion ⊇ is obvious. For the other inclusion let C be an (F p , F q )linear closed clonoid and let f be an n-ary function in C. In order to prove that is the constant n-ary function with value f (0 ). This implies the claim because the n-ary constant function with value f (0 ) is in C (p,q) (C [1] ) by Definition 1.1. We can see that f ′ is a 0-preserving function of C. The strategy is to interpolate f ′ in every line passing through the origin. To this end, let R = {L i | 1 ≤ i ≤ (q n − 1)/(q − 1) = s} be the set of all s distinct lines of the space F n q that pass through the origin, parametrized by the vectors l i ∈ F n q with i ∈ {1, . . . s} = I. For all i ∈ I, let f L i : F n q → F p be defined by: Since f ′ is 0-preserving we can write f ′ as: Then we prove by induction on the arity m that the function Case m = 1: if m = 1 then t 1 is a unary function of C [1] . Case m > 1: by the induction hypothesis we know that t m−1 ∈ C (p,q) (C [1] ). We define s m : F 2 q → F m q by s m (i, j) = (i, j, 0, . . . , 0). We denote by f sm(h,k) the Lagrange interpolation function of the point s m (h, k) (Definition 4.1). Let us define the function r : F n q → F p by: j∈Fq,a∈Fq\{0} f sm(j,ia −1 ) (x ), (6.1) for all x = (x 1 , . . . , x n ) ∈ F n q . Then we set s = i∈Fq g(i) = i∈Fq\{0} g(i). For all i ∈ F q \{0}, let ρ i , ψ i : F q × (F q \{0}) → F q × (F q \{0}) be the functions defined as ρ i (x, y) = (i + xy, y) and for all x ∈ F m q and i ∈ F q \{0}, as sums of three permutations of the same summands. Hence, we can write (6.1) as: for all x ∈ F m q , since g(0) = 0. Because of (6.1), we have r ∈ C (p,q) ({t m−1 }) ⊆ C (p,q) (C [1] ). Hence, (6.3) implies that qt m ∈ C (p,q) (C [1] ) and thus t m ∈ C (p,q) (C [1] ). This concludes the induction. Thus t n ∈ C (p,q) (C [1] ) and we can see that f L i (λl i ) = t n (λ(1, 0, . . . , 0)) = g(λ), for all λ ∈ F q , and f L i (x ) = t n (y ) = 0 for all x ∈ F n q \{λl i | λ ∈ F q } and y ∈ F n q \{λ(1, 0, . . . , 0) | λ ∈ F q }. By Lemma 6.1, , which concludes the proof.
The next two corollaries of Theorem 6.2 tell us that there are only finitely many distinct (F p , F q )-linear closed clonoids. Corollary 6.4. Let p and q be powers of distinct prime numbers. Then every (F p , F q )-linear closed clonoid has a set of finitely many unary functions as generators, and hence there are only finitely many distinct (F p , F q )-linear closed clonoids.

The lattice of all (F p , F q )-linear closed clonoids
In this section we investigate the structure of the lattice L(p, q) of all (F p , F q )linear closed clonoids through a characterization of their unary parts. We call the (F p , F q )-linear closed clonoids that are composed by only 0-preserving functions 0-preserving (F p , F q )-linear closed clonoids. We will see that there is an isomorphism between the sublattice of L(p, q) of the 0-preserving (F p , F q )-linear closed clonoids and the invariant subspaces lattice of a particular cyclic linear transformation A * (p, q) on the vector space F q−1 p . Let us start with the definition of a vector encoding of a unary function, concept that we need in order to see the connection between the (F p , F q )-linear closed clonoids and some invariant subspaces of F q p . Definition 7.1. Let F p and F q be finite fields and let f : F q → F p be a unary function. Let α be a generator of the multiplicative subgroup F × q of F q . We define the α-vector encoding of f as the vector v ∈ F q p such that: We denote by s α the function from F Fq p to F q p that sends a unary function f to its α-vector encoding.
Proposition 7.2. Let C be an (F p , F q )-linear closed clonoid, and let α be a generator of the multiplicative subgroup F × q of F q . Then the set S of all α-vector encodings of unary functions in C is a subspace of F q p and it satisfies: for all (x 0 , . . . , x q−1 ) ∈ S and k ∈ {1, . . . , q − 1}.
Proof. Let C be an (F p , F q )-linear closed clonoid. We can observe that the set S of the hypothesis is a subspace of F q p as a vector space by the first part of Definition 1.1. Moreover, (7.1) and (7.2) are a direct consequence of the second part of Definition 1.1, which guarantees that if f ∈ C [1] , then g : The closure under the linear transformation T : F p q → F p q defined as T ((x 0 , . . . , x q−1 )) = (x 0 , x 2 , . . . , x q−1 , x 1 ) is enough to describe the property to be closed with all the q − 1 linear transformations in (7.1) of Proposition 7.2. We denote by A(p, q) and C(p, q) respectively the matrices associated with the linear transformation T and with the linear transformation defined in (7.2). The minimal polynomial of T is x q−1 − 1. Let L(p, q) [1] = {C [1] | C ∈ L(p, q)}. We denote by e α : L(p, q) [1] → L(A(p, q), C(p, q)) the function that sends the set of unary functions of an (F p , F q )-linear closed clonoid to the set of their α-vector encodings.
Let us now characterize the closure properties of the unary part of an (F p , F q )linear closed clonoids.
Proof. Let X be as in the hypothesis, then we show that C (p,q) (X) = n∈N X n . The ⊇ inclusion is obvious by Definition 1.1. For the other inclusion we prove that S = n∈N X n is an (F p , F q )-linear closed clonoid.
Let f, g ∈ S [m] . Then there exist n 1 , n 2 ∈ N such that f ∈ X n 1 and g ∈ X n 2 . Let n 3 = max(n 1 , n 2 ), then f, g ∈ X n 3 and af + bg ∈ X n 3 +1 for all a, b ∈ F q . Let h ∈ S [m] . Then there exists k ∈ N such that h ∈ X k . Let A ∈ F m×s q . By definition of X k+1 , we have that the function g : (x 1 , . . . , x s ) → h(A · (x 1 , . . . , x s ) t ) is in X k+1 , and hence S is an (F p , F q )-linear closed clonoid. Furthermore S ⊇ X and thus S ⊇ C (p,q) (X), which is the smallest (F p , F q )-linear closed clonoid containing X. and for all f ∈ D and α ∈ F q , the function g α : x → f (αx) is in D. Then for all h ∈ C (p,q) (D) [m] , m ∈ N, and c ∈ F m q , the function l : Proof. Let D be as in the hypothesis and let S = C (p,q) (D). From Lemma 7.3, S = n∈N X n where: We prove by induction on n that for all m ∈ N, h ∈ X

[m]
n , and c ∈ F m q , l : x → h(xc) is in D, which implies that for all h ∈ C (p,q) (D) [m] , m ∈ N, and c ∈ F m q , the function l : x → h(xc) is in D.
Case n = 0: let g ∈ X 0 = D. Then the claim holds for the closure properties of D.
Case n > 0: suppose that the claim holds for n − 1. Let m ∈ N, f, g ∈ X [m] n−1 , and a, b ∈ F p . Then, for all c ∈ F m q the function t : x → (af + bg)(xc) = af (xc) + bg(xc) is in D, as a linear combination of functions in D. Furthermore, let k ∈ N and h ∈ X n . It is easy to see that n∈N X [1] n = C (p,q) (D) [1] . The ⊆ is obvious. For the other inclusion let f ∈ C (p,q) (D) [1] . Then, by Lemma 7.3, there exists n ′ ∈ N such that f ∈ X n ′ . But f is unary and so f ∈ X Let us now prove a theorem that restricts our research to only the 0-preserving unary parts of the (F p , F q )-liner closed clonoids.
Theorem 7.5. Let p and q be powers of distinct prime numbers. Then the lattice of all (F p , F q )-linear closed clonoids L(p, q) is isomorphic to the lattice L(A(p, q), C(p, q)) of all (A(p, q), C(p, q))-invariant subspaces of F q p . Proof. Let α be a generator of F × q . Let π : L(p, q) → L(p, q) [1] be such that π(C) = C [1] for all C ∈ L(a, b). We define the function γ : L(p, q) :→ L(A(p, q), C(p, q)) such that γ := e α • π. From Theorem 6.3 we have that π is bijective. Let ψ : L(A(p, q), C(p, q)) → L(p, q) [1] be defined by: . Let us show that ψ is well-defined. Let V ∈ L(A(p, q), C(p, q)). We show that the set of functions ψ(V ) respects the hypothesis of Lemma 7.4. Indeed, let f, g ∈ ψ(V ) and let a, b ∈ F p . Then, there exist v , w ∈ V such that s α (f ) = v and s α (g) = w . Thus, s α (af + bg) = av + bw ∈ V . Moreover, let c ∈ F q \{0}.
Furthermore, it is easy to see that π, e α , π −1 , e −1 α = ψ are monotone functions with respect to the set inclusion and hence the compositions γ = e α • π and γ −1 = π −1 • e −1 α are both monotone. Thus, γ is a bijective monotone function with monotone inverse, which implies that γ is a lattice isomorphism.
Theorem 7.6. Let p and q be two powers of distinct primes. Then the lattice L(A(p, q), C(p, q)) of all (A(p, q), C(p, q))-invariant subspaces of F q p is isomorphic to the direct product of the two-element lattice and the lattice of all 0-preserving (A(p, q), C(p, q))-invariant subspaces of F q p . Proof. Let V 0 , V 1 ⊆ F q p be defined by: It is clear that V 0 and V 1 are (A(p, q), C(p, q))-invariant subspaces of F q p and that the lattices L 0 and L 1 respectively of the (A(p, q), C(p, q))-invariant subspaces of V 0 and V 1 are isomorphic to L(A(p, q), C(p, q)) 0 and to 2. Moreover, V 0 ∨V 1 = 1 and V 0 ∩ V 1 = 0. Let W be an (A(p, q), C(p, q))-invariant subspace. Then we have that either W ≤ V 0 or W ≥ V 1 . Next, we show that using the modular law. Thus, L(A(p, q), C(p, q)) is isomorphic to 2 × L(A(p, q), C(p, q)) 0 .
The theorem describes the structure of the lattice of all (F p , F q )-linear closed clonoids in case p and q are powers of distinct primes. In the figure below we draw a scheme of the lattice of all (F p , F q )-linear closed clonoids. On the right hand side we have the 0-preserving part and on the left, the part with constants. With 1, 0 P , C, {0} we the denote respectively the (F p , F q )-linear closed clonoids of all functions, of all 0-preserving functions, of all constants, and of the zero constants.
The next step will be to describe the lattice L(A(p, q), C(p, q)) 0 of all 0-preserving (A(p, q), C(p, q))-invariant subspaces of F q p . From now on we will denote by A * (p, q) the restriction of A(p, q) formed by (A(p, q)) ij with i, j ∈ {2, . . . , q}. We can easily see that L(A(p, q), C(p, q)) 0 is isomorphic to L(A * (p, q)) the lattice of all A * (p, q)-invariant subspaces of F q−1 p , with the isomorphism π >2 : L(A(p, q), C(p , q)) 0 → L(A * (p, q)) such that: for all C ∈ L(A(p, q), C(p, q)) 0 . Thus, the last step is to characterize the lattice of the A * (p, q)-invariant subspaces of F q−1 p . The lattice of invariant subspaces under a linear transformation on finitedimensional vector spaces was characterized in [BF67]. We can see that A * (p, q) has minimal polynomial g = x q−1 −1. Let g = s i=1 p k i i be the prime factorization of g over F p [x ]. We define V i = ker(p i (A(p, q)) k i ), and A * (p, q) i = A * (p, q)| V i . We know from [BF67] that with the prime factorization of g we can split our vector space F q−1 p into its primary decomposition: [BF67] the lattice L(A * (p, q)) of the A * (p, q)-invariant subspaces of F q−1 p , is: We can observe that F p−1 q is an A * (p, q)-cyclic space generated by (1, 0, . . . , 0).
With these tools we are now ready to prove Theorem 1.2 and 1.3.
Proof of Theorem 1.3. Let p and q be powers of distinct primes and let n i=1 p k i i be the prime factorization of the polynomial g = x q−1 − 1 in F p [x]. First we know from Theorem 7.5 and Theorem 7.6 that L(p, q) ∼ = L(A(p, q), C(p, q)) ∼ = 2 × L(A(p, q), C(p, q)) 0 . Thus we can restrict to study the lattice L(A(p, q), C(p, q)) 0 ∼ = L(A * (p, q)) of the 0-preserving (A(p, q), C(p, q))-invariant subspaces. Moreover we know that A * (p, q) has g as minimal polynomial. So let V i be the ith primary component and let (A * (p, q)) i be the ith restriction of A * (p, q) with minimal polynomial p k i i , for i = 1, . . . , n. Then we know that F q−1 p is (A * (p, q))-cyclic with (1, 0, . . . , 0) as (A * (p, q))-cyclic vector. Hence also V i is (A * (p, q))-cyclic, for i = 1, . . . , n, as subspace of an (A * (p, q))-cyclic space. From [BF67, Lemma 2], the lattice of all A * (p, q) i -invariant subspaces of V i is isomorphic to the chain with k i + 1 elements. Thus, from (7.4) we have that: L(p, q) ∼ = 2 × n i=1 C k i +1 and the claim holds.
With this theorem we have completely characterized the structure of the lattice L(p, q) using the prime factorization of the polynomial x q−1 − 1, which can be easily computed. We conclude our investigation with a corollary that shows how the lattice of all (F p , F q )-linear closed clonoids is structured.
Corollary 7.7. Let p and q be powers of distinct primes. Then the lattice L(p, q) of the (F p , F q )-linear closed clonoids is a distributive lattice.
Proof. It follows from Theorem 1.3 that L(p, q) is a direct product of chains and hence is distributive.