aa r X i v : . [ m a t h . R A ] M a y CLONES CONTAINING THE MAL’CEV OPERATION OF Z pq STEFANO FIORAVANTI
Abstract.
We investigate finitary functions from Z pq to Z pq for two distinctprime numbers p and q . We show that the lattice of all clones on the set Z pq which contain the addition of Z pq is finite. We provide an upper bound for thecardinality of this lattice through an injective function to the direct productof the lattice of all ( Z p , Z q )-linearly closed clonoids to the p + 1 power andthe lattice of all ( Z q , Z p )-linearly closed clonoids to the q + 1 power. Theselattices are studied in [Fio19] and there we can find the exact cardinality ofthem. Furthermore, we prove that these clones can be generated by a set offunctions of arity at most max ( { p, q } ). Introduction
The investigation of the lattice of all clones on a set A has been a fecund fieldof research in general algebra with results such as Emil Post’s characterization ofthe lattice of all clones on a two-element set [Pos41]. This branch was developedfurther, e. g., in [Ros69, PK79, Sze86] and starting from [KBJ05], clones are usedto study the complexity of certain constraint satisfaction problems (CSPs).The aim of this paper is to describe the lattice of those clones on the set Z pq that contain the operation of addition of Z pq , with p and q distinct primes. Thuswe want to study the part of the lattice of all clones on Z pq which is above theclone of all linear mappings.In [Idz99] P. Idziak characterized the number of polynomial Mal’cev clones(clones containing the constants and a Mal’cev term) on a finite set A , which isfinite if and only if | A | ≤
3. In [Bul02] A. Bulatov shows a full characterizationof all infinitely many polynomial clones on the sets Z p × Z p and Z p that contain+, where p is a prime. Moreover, a description of polynomial clones on Z pq containing the sum for distinct primes p and q is given in [AM07] and polynomialclones containing + on Z n , for n squarefree, are described in [May08].In [Kre19] S. Kreinecker proved that there are infinitely many non-finitelygenerated clones above Clo( h Z p × Z p , + i ) for a prime p > F p , F q )-linearly closedclonoid as defined in [Fio19, Definition 1 . Date : June 2, 2020.1991
Mathematics Subject Classification.
Key words and phrases.
Clonoids, Clones.Supported by the Austrian Science Fund (FWF):P29931.
Definition 1.1.
Let p and q be powers of different primes, and let F p and F q betwo fields of orders p and q . An ( F p , F q ) -linearly closed clonoid is a non-emptysubset C of S n ∈ N F F nq 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 × nq : g : ( x , . . . , x n ) f ( A · ( x , . . . , x n ) t ) is in C [ n ] .In [Fio19, Thoerems 1 . .
3] we can find a complete description of thelattice of all ( F p , F q )-linearly closed clonoids with p and q powers of distinctprimes.In Section 4 we show an embedding of the lattice of all ( Z p , Z q )-linearly closedclonoids into the lattice of all clones above Clo( h Z pq , + i ), where p and q aredistinct primes. Theorem 1.2.
Let p and q be two distinct prime numbers. Then the lattice ofall ( Z p , Z q ) -linearly closed clonoids is embedded in the lattice of all clones above Clo( h Z pq , + i ) . In Section 5 we investigate the part of the lattice of all clones above Clo( h Z pq , + i )which is composed by all clones that can be split in two components over Z p andover Z q . These are the clones which preserve both π and π . We will show acharacterization of these clones that can be proven also using [AM15, Lemma6 . Theorem 1.3.
Let p and q be distinct prime numbers. Then there is an isomor-phism between the lattice of all clones above Clo( h Z pq , + i ) which preserve { π , π } and the direct product of the lattices of all clones above Clo( h Z p , + i ) and of allclones above Clo( h Z q , + i ) . In Section 6 we can find the main results of this paper that regard the cardi-nality of the lattice of all clones on Z pq that contain all linear mappings. Theorem 1.4.
Let p and q be distinct prime numbers and let Clo L ( h Z pq , + i ) be the lattice of all clones containing Clo( h Z pq , + i ) . Then there is an injectivefunction from Clo L ( h Z pq , + i ) to the direct product of the lattice of all ( Z p , Z q ) -linearly closed clonoids, L ( Z p , Z q ) , to the p +1 power and the lattice of all ( Z q , Z p ) -linearly closed clonoids, L ( Z q , Z p ) , to the q + 1 power, i. e: Clo L ( h Z pq , + i ) ֒ → L ( Z p , Z q ) p +1 × L ( Z q , Z p ) q +1 . Furthermore, from this result we can obtain a bound for the number of cloneson Z pq that contain Clo( h Z pq , + i ). Corollary 1.5.
Let p and q be distinct prime numbers. Let Q ni =1 p k i i and Q si =1 r d i i be the factorizations of respectively g p = x q − − in Z p [ x ] and g q = x p − − in LONES ON Z pq Z q [ x ] into their irreducible divisors. Then the number k of clones containing Clo( h Z pq , + i ) is bounded by: (1.1)2( n Y i =1 ( k i + 1) + s Y i =1 ( d i + 1)) − ≤ k ≤ p + q +2 n Y i =1 ( k i + 1) p +1 s Y i =1 ( d i + 1) q +1 ≤ qp + q + p . This theorem extends the finiteness result in [AM07] to clones on Z pq whichdo not necessarily contain constants, with p and q distinct primes. The mainingredient we used is [Fio19, Theorem 1 . F p , F q )-linearly closed clonoids for all p and q powers of distinctprimes.We can also use the proof of Theorem 1.4 to find a concrete bound on the arityof the generators of clones containing Clo( h Z pq , + i ). Corollary 1.6.
Let p and q be distinct prime numbers. Then the clones con-taining Clo( h Z pq , + i ) can be generated by a set of functions of arity at most max ( { p, q } ) . This corollary provides a bound for the arity of the generators of a clone con-taining Clo( h Z pq , + i ) which is max ( { p, q } ) and gives a rather low and unexpectedbound for the arity of the functions that really determine the clones.Furthermore, we use this result to find a description of some parts of the latticeof all clones above Clo( h Z pq , + i ). We denote by π , π the kernels of the two binaryprojections from Z p × Z q to respectively Z p and Z q . Theorem 1.7.
Let p and q be distinct prime numbers. Then there is an injectivefunction from the lattice of all clones above Clo( h Z pq , + i ) that preserve π and [ π , π ] = 0 to the direct product of the lattice of all clones above Clo( h Z p , + i ) andthe square of the lattice of all ( Z q , Z p ) -linearly closed clonoids. Preliminaries and notation
We use boldface letters for vectors, e. g., u = ( u , . . . , u n ) for some n ∈ N .Moreover, we will use h v , u i for the scalar product of the vectors v and u . Let A be a set and let 0 A ∈ A . We denote by A n a constant 0 A vector of length n .We denote by [ n ] the set { i ∈ N | ≤ i ≤ n } and by [ n ] the set [ n ] ∪ { } .Moreover we denote by N the set N ∪ { } . Let x ∈ Z np and let a ∈ [ p − n .Then we denote by x a the product Q ni =1 ( x ) ( a ) i i .From now one we will consider the group Z p × Z q instead of Z pq . We can seethat the two groups are isomorphic and thus equivalent for our purpose. Wesee that the congruence lattice of Z p × Z q is the square lattice with the fourcongruences { , , π , π } .The reason why we consider Z p × Z q instead of Z pq is that we want to distinguishthe component Z p from Z q when we consider the domain or the codomain of a STEFANO FIORAVANTI finitary function from Z p × Z q to itself. Moreover, we consider Z np × Z nq instead of( Z p × Z q ) n as domain of the functions we want to study. Let f : Z np × Z nq → Z p × Z q .We denote by f p : Z np × Z nq → Z p the function π Z p × Z q p ◦ f and by f q : Z np × Z nq → Z q the function π Z p × Z q q ◦ f , where ◦ is the symbol for the composition of functionsand π Z p × Z q i : Z p × Z p → Z i is the projection over Z i , with i ∈ { p, q } . Given avector of variables x = ( x , . . . , x n ) and m ∈ Z n we write x m for Q ni =1 x m i i .Let S be a set of finitary functions from a set A to itself . We denote by Clg( S )the clone generated by S on A . Let p and q be powers of distinct primes. We writeCig( F ) for the ( F p , F q )-linearly closed clonoid generated by a set of functions F ,as defined in [Fio19]. 3. Facts about clones
Let n ∈ N . We denote by Clo L ( h Z n , + i ) the lattice of all clones containingClo( h Z n , + i ).In [AM07] we can find a complete description of the polynomial clones (clonescontaining all constants) which contain Clo( h Z pq , + i ). In figure 1 we can see thelattice of all 17 distinct polynomial clones containing Clo( h Z pq , + i ), realized byP. Mayr.Our goal is to describe also those clones that do not necessary contain con-stants. Indeed, [AM07, Theorem 1 .
1] holds only for polynomial clones, and doesnot hold for clones that do not contain all the constants.Let us now show some basic facts about finitary functions from Z pq to Z pq starting from the definition of affine map. Definition 3.1.
Let G be a group. Then an affine map of G is an n -ary function f which satisfies:for all x , y , z ∈ G n : f ( x − y + z ) = f ( x ) − f ( y ) + f ( z ). Definition 3.2.
Let n ∈ N and let f : Z np × Z nq → Z p × Z q be a function. Then f is an affine map in the first (second) component if satisfies respectively:for all x , x , x ∈ Z np and y ∈ Z q : f ( x − x + x , y ) = f ( x , y ) − f ( x , y ) + f ( x , y )and for all x ∈ Z q and y , y , y ∈ Z np : f ( x , y − y + y ) = f ( x , y ) − f ( x , y ) + f ( x , y )for the second component. Remark 3.3.
It is a well-known fact that every finite field is polynomially com-plete. Thus for all f : F np → F p , there exists a sequence { a m } m ∈ [ p − n ⊆ F np suchthat for all x ∈ F np , f satisfies:(3.1) f ( x ) = X m ∈ [ p − n a m x m . LONES ON Z pq (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19)(cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) ②②②②②②②②②②②②②②② ❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊ (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19)(cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) ✭✭✭✭✭✭✭ rrrr ☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎ ✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴ (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19)(cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) ♦♦♦♦ ✯✯✯✯✯ ✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎ ✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾ (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19)(cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) ✭✭✭✭✭✭✭ rrrr ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19)(cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) ✭✭✭✭✭✭✭ ∗ rrrr ✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍ (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19)(cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) ♦♦♦♦ ∗ ✯✯✯✯✯ ✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴ (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19)(cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) ♦♦♦♦ ✯✯✯✯✯ ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19)(cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19)(cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) ✭✭✭✭✭✭✭ rrrr rrrr ✭✭✭✭✭✭✭ ✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎ ✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴ (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19)(cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) ✭✭✭✭✭✭✭ ∗ rrrr (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19)(cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) ✭✭✭✭✭✭✭ † rrrr ✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆ (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19)(cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) ♦♦♦♦ † ✯✯✯✯✯ ✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾ (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19)(cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) ♦♦♦♦ ∗ ✯✯✯✯✯ (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19)(cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) ✭✭✭✭✭✭✭ † rrrr ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19)(cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19)(cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) ✭✭✭✭✭✭✭ rrrr rrrr ✭✭✭✭✭✭✭ ❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀ (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19)(cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19)(cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) ✭✭✭✭✭✭✭ rrrr rrrr ✭✭✭✭✭✭✭ ✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄ (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19)(cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) ♦♦♦♦ † ✯✯✯✯✯ rrrrrrrrrrrrrrrrrrrrrrrrrrrrr (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19)(cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19)(cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) ✭✭✭✭✭✭✭ rrrr rrrr ✭✭✭✭✭✭✭ Figure 1.
Polynomial clones containing Clo( h Z pq , + i ): Each clone C is represented by its labelled congruence lattice. Simple factorsare labelled 2 if they are abelian and 3 otherwise. A minimal factorwhich is labelled 2 † is central; if it is labelled 2*, it is not central.(Picture and text realized by P. Mayr in [AM07]). STEFANO FIORAVANTI
In the following remark we see how the affine maps from Z np to Z p are charac-terized. Remark 3.4.
Let f : Z np → Z p be an n -ary function. Then f is affine if and onlyif there exist b ∈ Z np and c ∈ Z p such that for all x ∈ Z np (3.2) f ( x ) = h b , x i + c. This holds because f is affine if and only if for all x , y ∈ Z np , f ( x + y ) = f ( x ) + f ( y ) + f ( n ) if and only if f = f ′ + c , where f ′ is a homomorphism from Z np to Z p and c is a constant. Clearly, f ′ is a homomorphism from Z np to Z p ifand only if there exists b ∈ Z np such that for all x ∈ Z np (3.3) f ′ ( x ) = h b , x i , since every homomorphism from Z p to Z p is a linear mapping.Let p and q be distinct prime numbers. Let f : Z np → Z q be an n -ary function.With the same argument we can see that f is affine if and only if f is constant,since the only homomorphism from Z p to Z q is constant.Next we introduce the relation ρ ( α, β, δ, m ) following [AM07, Definition 2 . Definition 3.5.
Let A be an algebra, m : A → A , and α, β, γ ∈ Con( A ). Thenwe define the 4-ary relation ρ ( α, β, γ, m ) by: ρ ( α, β, γ, m ) := { ( a, b, c, d ) ∈ A | a α b, b β c, m ( a, b, c ) γ d } .A Mal’cev polynomial of A is a Mal’cev operation that lies in Pol ( A ). Let A be an algebra with a Mal’cev term m and let f ∈ Clo( A ). In [AM07, Proposition2 .
3] and in many other sources we can find that for all α, β, γ ∈ Con( A ) we havethat [ α, β ] ≤ γ if and only if α centralizes β modulo γ .Let f : A n → A be an n -ary function. We say that f preserves R ∈ A m if( f (( a ) , . . . , ( a n ) ) , . . . , f (( a ) m , . . . , ( a n ) m )) ∈ R for all a , . . . , a n ∈ R . Nowwe present a result in [AM07, Lemma 2 . ρ and the centralizer relation. Lemma 3.6.
Let A be an algebra in a congruence permutable variety, let m be a Malcev polynomial on A , and α, β, γ ∈ Con( A ) . Then the following areequivalent: (1) every f ∈ Pol( A ) preserves ρ ( α, β, γ, m ) . (2) α centralizes β modulo γ . Let A be an algebra in a congruence permutable variety, let m be a Malcevpolynomial on A , and let α, β, γ ∈ Con( A ). Let f : A n → A be an n -ary function.We say that f preserves [ α, β ] ≤ γ if f preserves ρ ( α, β, γ, m ).From now on we will fix the algebra, Z pq , and the Mal’cev term x − y + z .Hence, for the sake of simplicity, we will consider ρ as a ternary relation fixingthe Mal’cev term of Z pq as x − y + z .With the following Lemma we show some properties of functions on Z pq thatpreserve certain commutator relations. LONES ON Z pq Lemma 3.7.
Let p and q be different prime numbers. Then for all n ∈ N and f : Z np × Z nq → Z p × Z q the following hold: (1) f preserves π if and only if there exist f p : Z np → Z p and f q : Z np × Z nq → Z q such that f satisfies f ( x , y ) = ( f p ( x ) , f q ( x , y )) for all ( x , y ) ∈ Z np × Z nq ; (2) f preserves π if and only if there exist f p : Z np × Z nq → Z p and f q : Z nq → Z q such that f satisfies f ( x , y ) = ( f p ( x , y ) , f q ( y )) for all ( x , y ) ∈ Z np × Z nq ; (3) suppose that f preserves π . Then f is affine in the second component ifand only if f preserves [ π , π ] = 0 ; (4) suppose that f preserves π . Then f is affine in the first component ifand only if f preserves [ π , π ] = 0 ; (5) suppose that f preserves π . Then f q |{ n } × Z nq is an affine function from Z nq to Z q if and only if f preserves [1 , ≤ π ; (6) suppose that f preserves π . Then f p | Z np × { n } is an affine function from Z np to Z p if and only if f preserves [1 , ≤ π .Proof. Let us prove (1). Let n ∈ N and let f : Z np × Z nq → Z p × Z q . Then f preserves π if and only if for all (( x , y ) , ( x , y )) ∈ ( Z p × Z q ) whenever x = x then π Z p × Z q p ◦ f ( x , y ) = π Z p × Z q p ◦ f ( x , y ), where π Z p × Z q p : Z p × Z q → Z p is theprojection over Z p . This holds if and only if f p = π Z p × Z q p ◦ f depends on only thevariables from Z p . Thus (1) holds. The proof of item (2) is symmetric to the oneof item (1).Next we prove (3). Let n ∈ N and let f : Z np × Z nq → Z p × Z q . By defi-nition f preserves [ π , π ] = 0 if and only if f preserves ρ ( π , π , f preserves ρ ( π , π ,
0) if and only if for all x , x , x , x ∈ Z np and y , y , y , y ∈ Z nq , x = x = x = x and y − y + y = y then:(3.4) f ( x , y − y + y ) = f ( x , y ) − f ( x , y ) + f ( x , y ) . Clearly, (3.4) holds if and only if f is affine in the second component (Definition3.2). The proof of item (4) is symmetric to the one of item (3).Let us prove (5). Let n ∈ N and let f : Z np × Z nq → Z p × Z q . By definition f preserves [1 , ≤ π if and only if f preserves ρ (1 , , π ). Thus, by Definition 3.5, f preserves ρ (1 , , π ) if and only if for all x , x , x , x ∈ Z np and y , y , y , y ∈ Z nq if y − y + y = y then(3.5) f ( x , y − y + y ) π f ( x , y ) − f ( x , y ) + f ( x , y ) . By hypothesis f preserves π and thus, by item (2), (3.5) holds if and only if f q = π Z p × Z q q ◦ f satisfies:(3.6) f q ( y − y + y ) = f q ( y ) − f q ( y ) + f q ( y ) , for all y , y , y ∈ Z nq . Hence this holds if and only if f q |{ n } × Z nq is an affinefunction. The proof of item (6) is symmetric to the one of item (5). STEFANO FIORAVANTI (cid:3)
In the case p , . . . , p m are distinct prime numbers we can see that we can splita function f : Q mi =1 Z np i → Q mi =1 Z p i in f = P mi =1 f i , where f i = Q j ∈ [ m ] \{ i } p p i − j f .This implies, for example, that we can prove the following remark. Remark 3.8.
Let p · · · p m = s be a product of distinct prime numbers and let C be a clone containing Clo( h Z s , + i ). Then for all k ∈ N and ( a , . . . , a m ) ∈ Q i =1 Z kp i , h ( a ,..., a m ) : Q mi =1 Z kp i → Q mi =1 Z p i defined by h ( a ,..., a m ) : ( x , . . . , x m ) ( h a , x i , . . . , h a m , x m i ) is in C . Proof.
Let p · · · p m = s be a product of distinct prime numbers and let C be a clone containing Clo( h Z s , + i ). Let h p i : Q i =1 Z p i → Q i =1 Z p i be suchthat h p i : ( x , . . . , x m ) (0 Z p , . . . , . . . , Z pi − , x i , Z pi +1 , . . . , Z pm ). Then h =( Q j ∈ [ m ] \{ i } p j ) p i − π , where π is the unary projection of Clo( h Z pq , + i ). Thus π and the sum generate the function h p i for all i ∈ [ m ]. Clearly, with { h p i } i ∈ [ m ] ,the sum, and the projections we can generate every other linear combination ofthe variables in the m components. (cid:3) Let A be a set and let F p be a field of order p . With the following lemma weshow that every function from F np × A s to F p can be seen as the induced functionof a polynomial of R [ X ], where R = F A s p . This easy fact will be often used later. Lemma 3.9.
Let A be a set and let F p be a field of order p . Then for everyfunction f from F np × A s to F p there exists a sequence of functions { f m } m ∈ [ p − n from A s to F p such that f satisfies for all x ∈ F np , y ∈ A s : (3.7) f ( x , y ) = X m ∈ [ p − n f m ( y ) x m . Proof.
Let f be a function from F np × A s to F p and let us fix the variables over A s as a vector a ∈ A s and let f a : F np → F p be defined by f a : x f ( x , a ).By Remark 3.3, we have that for every a ∈ A s there exists { c ( m , a ) } m ∈ [ p − n suchthat for all x ∈ F np f satisfies:(3.8) f ( x , a ) = f a ( x ) = X m ∈ [ p − n c ( m , a ) x m . Thus we can see that for all x ∈ Z np , y ∈ Z np :(3.9) f ( x , y ) = X a ∈ A s χ a ( y ) X m ∈ [ p − n c ( m , a ) x m , where χ a ( a ) = 1 and χ a ( y ) = 0, for all y ∈ A s \{ a } . Thus for all m ∈ [ p − n we define f m : A s → F p by: LONES ON Z pq (3.10) f m := X a ∈ A s c ( m , a ) χ a . Thus the claim holds. (cid:3)
The previous lemma in our setting implies the following.
Lemma 3.10.
Let p and q be prime numbers. Then for every function f from Z np × Z nq to Z p × Z q there exist two sequences of functions { f m } m ∈ [ p − n from Z nq to Z p and { f h } h ∈ [ q − n from Z np to Z q such that f satisfies for all x ∈ Z np , y ∈ Z nq : (3.11) f ( x , y ) = ( X m ∈ [ p − n f m ( y ) x m , X h ∈ [ q − n f h ( x ) y h ) . Embedding of the Clonoids
The aim of this section is to prove that there exists an embedding of thelattice of all ( Z p , Z q )-linearly closed clonoids in the lattice of all clones containingClo( h Z pq , + i ), when p and q are distinct prime numbers.Let e : S n ∈ N Z Z nq p → S n ∈ N ( Z p × Z q ) Z np × Z nq be such that for all n ∈ N and for all f ∈ Z Z nq p , e ( f ) := g , where g : Z np × Z nq → Z p × Z q satisfies for all ( x , y ) ∈ Z np × Z nq , g ( x , y ) = ( f ( y ) , γ from the lattice of all ( Z p , Z q )-linearly closed clonoidsto the lattice of all clones containing Clo( h Z pq , + i ) such that for all C ∈ L ( Z p , Z q ):(4.1) γ ( C ) := { f | f : Z np × Z nq → Z p × Z q , ∃ ( g ∈ C, a ∈ Z np , b ∈ Z nq ) : f = e ( g )+ h ( a , b ) } , where h ( a , b ) : Z np × Z nq → Z p × Z q is defined in Remark 3.8.In order to prove Theorem 1.2 we first show a lemma. Lemma 4.1.
Let p and q be powers of distinct primes. Let X ⊆ S n ∈ N F F nq p . Then Cig( X ) = S n ∈ N X n where: X = XX n +1 = { af + bg | a, b ∈ F p , f, g ∈ X [ r ] n , r ∈ N } ∪ { g : ( x , . . . , x l ) f ( A · ( x , . . . , x l ) t ) | f ∈ X [ k ] n , A ∈ F k × lq } .Proof. Let X be as in the hypothesis. Then we show that Cig( X ) = S n ∈ N X n .The ⊇ inclusion is obvious by Definition 1.1. For the other inclusion we provethat S = S n ∈ N X n is an ( F p , F q )-linearly closed clonoid.Let f, g ∈ S [ r ] . Then there exist n , n ∈ N such that f ∈ X n and g ∈ X n .Let n = max ( { n , n } ) then f, g ∈ X n and af + bg ∈ X n +1 for all a, b ∈ F p .Furthermore, let h ∈ S [ k ] . Then there exists n ′ ∈ N such that h ∈ X n ′ . Let A ∈ F k × rq . By definition of X n ′ +1 , we have that the function g : x h ( A · x ),for all x ∈ F rq , is in X n ′ +1 , and hence S is an ( F p , F q )-linearly closed clonoid.Furthermore S ⊇ X and thus S = Cig( X ), which is the smallest ( F p , F q )-linearlyclosed clonoid containing X . Thus the claim holds. (cid:3) Proof of Theorem 1.2.
Let γ be the function defined in (4.1). First we show that γ is well-defined. Let C be a ( Z p , Z q )-linearly closed clonoid. Then γ ( C ) containsthe projections and the binary sum of Z p × Z q . Moreover, let f, f , . . . , f n ∈ γ ( C )be respectively an n -ary and n m -ary functions. Then there exist g f , g , . . . , g n ∈ C , a f ∈ Z np , b f ∈ Z nq , a , . . . , a n ∈ Z mp , and b , . . . , b n ∈ Z mq such that: f ( x , y ) = ( h a f , x i + g f ( y ) , h b f , y i ),for all x ∈ Z np , y ∈ Z nq , and for all 1 ≤ i ≤ n : f i ( x , y ) = ( h a i , x i + g i ( y ) , h b i , y i ),for all x ∈ Z mp , y ∈ Z mq . Then h = f ◦ ( f , . . . , f n ) can be written as: h ( x , y ) = ( h a h , x i + g h ( y ) , h b h , y i ),where for all j = 1 , . . . , m , ( a h ) j = P ni =1 ( a f ) i ( a i ) j , ( b h ) j = P ni =1 ( b f ) i ( b i ) j , and g h : Z mq → Z p is defined by g h ( y ) = h a f , d ( y ) i + g f ( h b , y i , . . . , h b n , y i ) with d ( y ) = ( g ( y ) , . . . , g n ( y )), for all y ∈ Z mq . We can see from Definition 1.1 that g h ∈ C . Thus γ ( C ) is closed under composition and γ is well-defined.Next we prove that γ is injective. Let C and D be two ( Z p , Z q )-linearly closedclonoids such that γ ( C ) = γ ( D ) and let g ∈ C be an n -ary function. Then let s : Z np × Z nq → Z p × Z q be such that e ( g ) = s . Then s is in γ ( C ) = γ ( D ).By definition of γ , this implies that e ( g ) = e ( g ′ ) + h ( a , b ) for some g ′ ∈ D and( a , b ) ∈ Z np × Z nq . The only possibility is that g = g ′ ∈ D and thus C ⊆ D . We canrepeat this argument for the other inclusion and hence γ is injective. Furthermore,we have that for all C, D ∈ L ( Z p , Z q ) clearly γ ( C ∩ D ) = γ ( C ) ∩ γ ( D ). We canobserve that γ is monotone, thus γ ( C ∨ D ) ⊇ γ ( C ) ∨ γ ( D ). For the other inclusionwe prove by induction on n that γ ( C ) ∨ γ ( D ) ⊇ e ( X n ), where C ∨ D = S n ∈ N X n with: X = C ∪ DX n +1 = { af + bg | a, b ∈ F p , f, g ∈ X [ m ] n , m ∈ N } ∪ { g : ( x , . . . , x s ) f ( A · ( x , . . . , x s ) t ) | f ∈ X [ m ] n , A ∈ Z m × sq , m, s ∈ N } .Base step n = 0: e ( C ∪ D ) = e ( C ) ∪ e ( D ) ⊆ γ ( C ) ∨ γ ( D ).Induction step n >
0: suppose that the claim holds for n −
1. Then let g ∈ e ( X n ). Hence there exists u ∈ X n such that e ( u ) = g and either u is alinear combination of functions in X n − or there exist f ∈ X [ m ] n − , A ∈ Z m × sq , and m, s ∈ N such that u : x f ( A · x t ). In both cases we have g ∈ Clg( e ( X n − )) ⊆ LONES ON Z pq γ ( C ) ∨ γ ( D ) and this concludes the induction proof. By Lemma 4.1, γ ( C ) ∨ γ ( D ) ⊇ e ( C ∨ D ).Let f ∈ γ ( C ∨ D ) [ n ] . Then there exist g ∈ C ∨ D , a ∈ Z np , and b ∈ Z nq such that f = e ( g ) + h ( a , b ) . By Remark 3.8, h ( a , b ) ∈ γ ( C ) ∨ γ ( D ), thus f ∈ γ ( C ) ∨ γ ( D )and γ ( C ) ∨ γ ( D ) = γ ( C ∨ D ). Hence γ is an embedding. (cid:3) Independent algebras
In this section we characterize all clones containing Clo( h Z pq , + i ) that preservethe whole congruence lattice of four elements. In this cases we see that we canactually split these clones on Z pq in two clones, one containing Clo( h Z p , + i ) andthe other containing Clo( h Z q , + i ). To this end, let us introduce the concept ofindependent algebras (see also [AM15]). Definition 5.1.
Two algebras A and B of the same variety V are independent if there exists a binary term in the language of V such that A | = t ( x, y ) ≈ x and B | = t ( x, y ) ≈ y .Now we prove a theorem that characterizes the clones containing Clo( h Z pq , + i )whose algebra can be split in a direct product of two independent algebras. Wedenote by Clo ⋄ ( h Z pq , + i ) the lattice of all clones containing Clo( h Z pq , + i ) whichpreserve the congruences { , π , π , } . We will present an independent proof ofTheorem 1.3 which also follows from [AM15, Lemma 6 . Proof of Theorem 1.3.
Let ρ : Clo ⋄ ( h Z pq , + i ) → Clo L ( h Z p , + i ) and ρ : Clo ⋄ ( h Z pq , + i ) → Clo L ( h Z q , + i ) be defined by: ρ ( C ) := { f | there exist n ∈ N , g ∈ C [ n ] : f = π Z p × Z q p ◦ g | Z np × { n }} ρ ( C ) := { f | there exist n ∈ N , g ∈ C [ n ] : f = π Z p × Z q q ◦ g | { n } × Z nq } , (5.1)where π Z p × Z q p , π Z p × Z q q are respectively the projections over Z p and over Z q . Let ρ :Clo( h Z pq , + i ) → Clo( h Z p , + i ) × Clo( h Z q , + i ) be defined by ρ : C ( ρ ( C ) , ρ ( C ))for all C ∈ Clo( h Z pq , + i ). First of all let us prove that ρ is well-defined. Tothis end, let C ∈ Clo( h Z pq , + i ). By (1) and (2) of Lemma 3.7, for all n ∈ N and f ∈ C [ n ] there exist f p : Z np → Z p and f q : Z nq → Z q such that for all( x , y ) ∈ Z np × Z nq , f ( x , y ) = ( f p ( x ) , f q ( y )). Furthermore, let n ∈ N . By Remark3.8, we have that the function h : Z np × Z nq → Z p × Z q defined by h : ( x , y ) (( x ) i , ( y ) i ) is in C and thus the projections π ni ∈ ρ ( C ), for all 1 ≤ i ≤ n . Let f ∈ ρ ( C ) [ n ] and g , . . . , g n ∈ ρ ( C ) [ m ] . Then there exist f ′ , g ′ , . . . , g ′ n ∈ C suchthat f = π Z p × Z q p ◦ f ′ | Z np × { n } and for all 1 ≤ i ≤ n , g i = π Z p × Z q p ◦ g ′ i | Z mp × { m } . Thus we have that f ′ ◦ ( g ′ , . . . , g ′ n ) ∈ C . Hence f ◦ ( g , . . . , g n ) = π Z p × Z q p ◦ f ′ ◦ ( g ′ , . . . , g ′ n ) | Z mp × { m } ∈ ρ ( C ). Furthermore, the binary sum of Z p is in C , since, by Remark 3.8, the function h : Z p × Z q → Z p × Z q defined by h : (( x , y ) , ( x , y )) ( x + x ,
0) is in C . Hence ρ ( C ) ∈ Clo( h Z p , + i ) and ρ is well-defined. Symmetrically, we can prove that ρ is well-defined.We define the function ψ : Clo L ( h Z p , + i ) × Clo L ( h Z q , + i ) → Clo ⋄ ( h Z pq , + i ) suchthat ψ ( C , C ) := { f | there exist n ∈ N , f ∈ C [ n ]1 , f ∈ C [ n ]2 : for all ( x , y ) ∈ Z np × Z nq , f ( x , y ) = ( f ( x ) , f ( y )) } .Clearly, ψ is well-defined. We prove that ρ ◦ ψ = Id Clo L ( h Z p , + i ) × Clo L ( h Z q , + i ) and ψ ◦ ρ = Id Clo ⋄ ( h Z pq , + i ) . First we can see that both ρ and ψ are monotone.For the first inequality let ( C , C ) ∈ Clo L ( h Z p , + i ) × Clo L ( h Z q , + i ). For themonotonicity of both the functions we have that ρ ◦ ψ (( C , C )) ⊇ ( C , C ).Let ( f , f ) ∈ ρ ◦ ψ (( C , C )) [ n ] . Then there exist g , g ∈ ψ (( C , C )) such that f = π Z p × Z q p ◦ g | Z np × { n } and f = π Z p × Z q q ◦ g | { n } × Z nq . Thus, by definitionof ψ , ( f , f ) ∈ ( C , C ) and the first inequality holds.For the second inequality let C ∈ Clo ⋄ ( h Z pq , + i ). For the monotonicity of both ψ and ρ we have that ψ ◦ ρ ( C ) ⊇ C . Let f ∈ ψ ◦ ρ ( C ). Then there exists( f , f ) ∈ ρ ( C ) such that for all ( x , y ) ∈ Z np × Z nq , f ( x , y ) = ( f ( x ) , f ( y )). Thenthere exist g , g ∈ C such that f = π Z p × Z q p ◦ g | Z np × { n } and f = π Z p × Z q q ◦ g |{ n } × Z nq . Thus f = q ( p − g + p ( q − g ∈ C and the second inequality holds.Thus ψ = ρ − and ρ are monotone bijective functions with monotone inverseand hence ρ is a lattice isomorphism. (cid:3) Corollary 5.2.
Let p and q be distinct prime numbers. Then there are ( n ( q ) +3)( n ( p )+3) many clones containing Clo( h Z pq , + i ) which preserve the congruences { , π , π , } , where n ( k ) is the number of divisors of k − .Proof. From Theorem 1.3 we obtain that the number of clones containing Clo( h Z pq , + i ) is the product of the number of clones containing Clo( h Z p , + i ) and the num-ber of clones containing Clo( h Z q , + i ). These numbers are determined in [Kre19,Cororllary 1 . p > p = 2. From these two results we havethat for every prime number p the clones containing Clo( h Z p , + i ) are n ( p ) + 3,where n ( k ) is the number of divisors of k − (cid:3) This complete the characterization of all clones containing Clo( h Z pq , + i ) whichbelong to products of independent algebras.6. A general bound
In the following section our goal is to determine a bound for the cardinality ofthe lattice of all clones containing Clo( h Z pq , + i ) and to find a bound for the arity ofsets of generators of these clones. Theorem 6.9 gives a complete list of generatorsfor a clone containing Clo( h Z pq , + i ) that explains the connection between clonoidsand clones in this case. The generators of Theorem 6.9 are substantially formedby a product of a unary function generating a ( Z p , Z q )-linearly closed clonoid and LONES ON Z pq a monomial generating a clone on Z p . This puts together the characterizationin [Kre19] and [Fio19, Theorems 1 .
2, 1 . h Z pq , + i ) are strictly characterized by the ( Z p , Z q )-linearly closed clonoids andthe ( Z q , Z p )-linearly closed clonoids.In this paper we have to deal with polynomials whose coefficients are finitaryfunctions from Z q to Z p . From now on let R [ X ] be a polynomial ring where R = Z np for some n ∈ N . The next step will be to generalize some results in[Kre19] about p -linearly closed clonoids to polynomials in R [ X ]. Let us startwith the notation.Following [Kre19] we denote by tD( h ) the total degree of a monomial h , whichis defined as the sum of the exponents. We denote by Var( h ) the set of variablesoccurring in h and with Var( f ) the set of variables occurring in monomials of f with non-zero coefficients. We denote by MInd( f ) the maximum of the indicesof variables in Var( f ). Let f ∈ R [ X ] be such that MInd( f ) = n and let x be avector composed by the variables in ( x , . . . , x n ). Then f can be written as:(6.1) f = X m ∈ N n r m x m , for some sequence { r m } m ∈ N n in R with only finitely many non-zero members andwhere x m = Q ni =1 x m i i We denote the set of all degrees of f by DEGS( f ) := { m | m ∈ N n , r m = 0 } with DEGS(0) := ∅ . We denote by MON( f ) the set of all monomials in f . Wedenote by TD( f ) := {h m , n i | m ∈ DEGS( f ) } .Next we introduce a notation for the composition of multivariate polynomials.Let m, n, h ∈ N , g, f , . . . , f h ∈ R [ X ], and let b = ( b , . . . , b h ) ∈ N h with 1 ≤ b < b < · · · < b h ≤ MInd( f ). Then we define g ◦ b ( f , . . . , f m ) by:(6.2) g ◦ b ( f , . . . , f h ) := g ( x , . . . , x b − , f , x b +1 , . . . , x b − , f , x b +1 , . . . ) . Let p be a prime and let f ∈ R [ X ], where R = Z np . Since later we want to intro-duce the induced function of a polynomial, in order to have a unique polynomialfor every induced function, we consider the ideal I generated by the polynomials x pi − x i ∈ R [ X ] in R [ X ], for every x i ∈ X . By [Eis95, Chapter 15 .
3] there is aunique reminder rem ( f ) of f with respect to I . This remainder has the propertythat the exponents of the variables are less or equal p −
1. Following [Kre19, Sec-tion 2], we denote by R ∗ [ X ] the set of all polynomials in R [ X ] with this property.We can observe that these polynomials form a set of representatives of the set ofall classes of the quotient R /I .Let f ∈ R ∗ [ X ] with MInd( f ) = n and let x be a vector composed by thevariables in ( x , . . . , x n ). Then f can be written as: (6.3) f = X m ∈ [ p − n r m x m , for some sequence { r m } m ∈ [ p − n in R . Definition 6.1.
Let p be a prime number and let R = Z np . An R - polynomiallinearly closed clonoid is a non-empty subset C of R [ X ] with the following prop-erties:(1) for all f ∈ C , g ∈ C , and a, b ∈ Z p af + bg ∈ C ;(2) for all s ∈ N , f ∈ C , l ≤ MInd( f ), and a ∈ Z sp : g = f ◦ l ( P si =1 ( a ) i x i ) is in C .Let S ⊆ R [ X ]. Then we denote by h S i R the R - polynomial linearly closedclonoid generated by S . We can see that R ∗ [ X ] forms an R -linearly closedclonoid.Let us now modify [Kre19, Lemmata 3 . .
9] to deal with R -polynomiallinearly closed clonoids. Since the statements of the following two lemmata arenot the same of [Kre19, Lemmata 3 . .
9] we show them with proofs thatare essentially the same of [Kre19].
Lemma 6.2.
Let d ∈ N , let r ∈ R and let g ∈ R [ X ] with max (TD( g )) ≤ d , and d DEGS( g ) . Then rx · · · x d ∈ h{ rx · · · x d + g }i R .Proof. If g = 0 we have nothing to show. Let g = 0 and let C := h{ rx · · · x d + g }i R . We prove that there exists g ′ ∈ R [ X ] with max (TD( g ′ )) ≤ d , d DEGS( g ′ ) and Var( g ′ ) ⊆ { x , . . . , x d } such that rx · · · x d + g ′ ∈ C .Case Var( g ) ⊆ { x , . . . , x d } : in this case the claim obviously holds.Case Var( g )
6⊆ { x , . . . , x d } : let B := Var( g ) \{ x , . . . , x d } . Let k ∈ N andlet b , . . . , b k ∈ N be such that b < · · · < b k and B = { x b , . . . , x b k } . We set g ′ ∈ R [ X ] such that g ′ := g ◦ ( b ,...,b k ) k . It follows that Var( g ′ ) ⊆ { x , . . . , x d } and we can observe that:(6.4) rx · · · x d + g ′ = ( rf x · · · x d + g ) ◦ ( b ,...,b k ) k ∈ C. Next we proceed by induction on the number of monomials of g ′ in order to showthat rx · · · x d ∈ h{ rx · · · x d + g ′ }i R ⊆ C .If g ′ has zero monomial then the claim obviously holds. Let us suppose thatthe claim holds for a number of monomials t ≥
0. Let the number of monomialsof g ′ be t + 1. We observe that there exists x l ∈ { x , . . . , x d } and a monomial m of g ′ such that x l does not appear in m . Thus we obtain:(6.5) rx · · · x d + g ′ − ( rx · · · x d + g ′ ) ◦ ( l ) rx · · · x d + g ′′ ∈ C. LONES ON Z pq Thus g ′′ := g ′ − g ′ ◦ ( l ) max (TD( g ′′ )) ≤ d , d DEGS( g ′′ ), Var( g ′′ ) ⊆ { x , . . . , x d } and g ′′ has fewer monomials than g ′ , since themonomial m is cancelled in g ′ − g ′ ◦ ( l )
0. By the induction hypothesis rx · · · x d ∈h{ rf x · · · x d + g ′′ }i R ⊆ h{ rf x · · · x d + g ′ }i R and the claim holds. (cid:3) With Lemma 6.2 we can now prove the following generalization of [Kre19,Lemma 3 .
9] with the same proof.
Lemma 6.3.
Let d ∈ N and let f be a polynomial in R ∗ [ X ] with d := max (TD( f )) .Let m ∈ MON( f ) be a monomial with coefficient r ∈ R and tD( m ) = d . Thenthe following hold: (1) if d = 0 , then r ∈ h{ f }i R ; (2) if d > , then rx . . . x d ∈ h{ f }i R .Proof. Let n = MInd( f ), let f = P m ∈ [ p − n r m x m ∈ R ∗ [ X ], C := h{ f }i R , let d := max (TD( f )), and let m ∈ MON( f ) with coefficient r ∈ R and tD( m ) = d .Case d = 0: then clearly r ∈ h{ f }i R .Case d >
0: let us fix an s ∈ DEGS( f ) with h s , n i = d and let r be thecoefficient of x s in f . Now we show by induction that for all m ∈ N with | Var( x s ) | ≤ m ≤ d there exists a monomial h := r x t ∈ R ∗ [ X ] with | Var( h ) | = m ,tD( h ) = d , and there exists g ∈ R ∗ [ X ] with max (TD( g )) ≤ d and t DEGS( g )such that h + g ∈ C .Base step m = | Var( x s ) | : in this case we see that we can take h = r x s and g = P m ∈ [ p − n \{ s } r m x m .Induction step: suppose that the claim holds for m < d . Then there existsa monomial h := r x t ∈ R ∗ [ X ] and g ′ ∈ R ∗ [ X ] with | Var( h ) | = m , tD( h ) = d , max (TD( g ′ )) ≤ d , and t DEGS( g ′ ) such that h + g ′ ∈ C . Then there exists j such that p − ≥ ( t ) j = u >
1, since m < d . Without loss of generality wesuppose that Var( h ) = { x , . . . , x m } . Let g ′′ := g ′ ◦ ( j ) ( x j + x m +1 ). We denote by y the vector of variables ( x , . . . , x m +1 ). Thus( h + g ′ ) ◦ ( j ) ( x j + x m +1 ) = r x t ◦ ( x j + x m +1 ) + g ′ ◦ ( j ) ( x j + x m +1 )= r ( u X k =0 (cid:18) uk (cid:19) x u − kj x km +1 ) · s Y i =1 ,i = j ( x ) ( t ) i i + g ′′ = r · ( t ) j · y (( t ) ,..., ( t ) j − , ( t ) j − , ( t ) j +1 ,..., ( t ) m , ++ r ( u X k =0 ,k =1 (cid:18) uk (cid:19) x u − kj x km +1 ) · Y i =1 ,i = j ( x ) ( t ) i i + g ′′ . (6.6)Let h := r · y (( t ) ,..., ( t ) j − , ( t ) j − , ( t ) j +1 ,..., ( t ) m , g = ( t ) − j r ( u X k =0 ,k =1 (cid:18) uk (cid:19) x u − kj x km +1 ) · Y i =1 ,i = j ( x ) ( t ) i i + ( t ) − j g ′′ . (6.7)Then h satisfies tD( h ) = d , | Var( h ) | = m +1. Furthermore, g satisfies max (TD( g )) ≤ d and (( t ) , . . . , ( t ) j − , ( t ) j − , ( t ) j +1 , . . . , ( t ) m , DEGS( g ). Thus h and g are the searched polynomials. This concludes the induction step.Thus, by Definition 6.1, rx . . . x d + g ′′′ ∈ C for some g ′′′ ∈ R ∗ [ X ] with max (TD( g ′′′ )) ≤ d . By Lemma 6.2 we have that rx . . . x d ∈ C and the claimholds. (cid:3) We are now ready to prove that an R -polynomial linearly closed clonoid gen-erated by an element of R ∗ [ X ] contains every monomial of every polynomial init. Lemma 6.4.
Let f ∈ R ∗ [ X ] be such that h = r m x m is a monomial of f . Then h ∈ h f i R .Proof. The proof is by induction on the number n of monomials in f .Base step n = 1: then clearly the claim holds.Induction step n >
0: suppose that the claim holds for every g with n − f be a polynomial with n monomials with non-zero coefficients.Let d = max (TD( f )) and let h be a monomial in f with degree d and coefficient r h . By Lemma 6.3, we have that either r h x · · · x d ∈ h f i R if d > r h ∈ h f i p if d = 0. Clearly, this yields h ∈ h f i R . From the induction hypothesis we havethat all n − f − h are in h f − h i R ⊆ h f i R . Thus all monomialsof f are in h f i R . (cid:3) For every polynomial f ∈ ( Z Z nq p ) ∗ [ X ] with MInd( f ) = k of the form f = P m ∈ [ p − k r m x m we define its s -ary induced function f [ s ] : Z sp × Z sq → Z p × Z q by: ( x , y ) ( P m ∈ [ p − k r m (( y ) , . . . , ( y ) n ) Q ki =1 ( x ) ( m ) i i , s ≥ k, n . From now on, when not specified, s = max ( { MInd( f ) , n } ). Nextwe show a lemma that connects the monomials of an R -polynomial linearly closedclonoid to functions of a clones on Z pq . Lemma 6.5.
Let p and q be distinct prime numbers. Let R = Z Z nq p and let h, h ∈ R ∗ [ X ] with h ∈ h h i R . Then h ∈ Clg( { h } ) .Proof. We know that every clone C containing Clo( h Z pq , + i ) is closed under com-position and, by Remark 3.8, contains every linear mapping h ( a , b ) with a ∈ Z np and b ∈ Z nq . Then it is clear that if a monomial h can be generated from h withitem (1) or (2) of Definition 6.1, then the induced monomial h generates h in LONES ON Z pq a clone containing Clo( h Z pq , + i ), simply composing h with the linear mappings h ( a , b ) from the right, with a ∈ Z np and b ∈ Z nq , and with the sum from theleft. (cid:3) In order to show the following lemma we want to make an easy observation.Let h = r x d ∈ R ∗ [ X ] be a monomial, where with x d we denote the monomial x · · · x d and let d ∈ N \{ } . Furthermore, let R = Z Z nq p . Relabelling the vari-ables and composing r x d with linear mappings and itself we can observe thatClg( { r x d } ) ⊇ { r x d , r x d + d − , . . . , r k x d +( k − d − , . . . } . We have that r p = r modulo p , thus some of these monomials have the same coefficients. Moreover, ifwe substitute x to p variables we obtain that p − x pi and x i induce the same function and have the same representative in R ∗ [ X ].With these easy observations we can prove the following. Lemma 6.6.
Let p and q be distinct prime numbers. Let d ∈ N \{ } . Then forall k, l ∈ N , for all r ∈ Z Z lq p , and for all m ∈ [ p − k \{ k } with tD( x m ) = u congruent to d modulo p − it follows that: r x m ∈ Clg( { rx · · · x d } ) .Proof. Let p be a prime number and let d ∈ N \{ } . Clearly, r x m ∈ h rx · · · x d ′ i R ,where d ′ = tD( x m ), hence, by Lemma 6.5, we have that r x m ∈ Clg( { rx · · · x d ′ } ).Next we prove that Clg( { rx · · · x d ′ } ) ⊆ Clg( { rx · · · x d } ). If d ′ < d then, inorder to generate rx · · · x d ′ we compose rx · · · x d with a linear mapping that isthe first projection over Z p and is the identity over Z q . In this way we collapsethe variables { x d ′ +1 , . . . , x d } , since x p = x modulo p and d ′ = d modulo p − d ′ > d , then we compose rx · · · x d to itself relabelling the variables and wegenerate the induced monomials { r x d , r x d + d − , . . . , r k x d +( k − d − , . . . } . Weknow that r p = r and thus in particular we generate { r x d , r x d +( p − d , . . . ,r x d + c ( p − d , . . . } for all c ∈ N . For a suitable k ∈ N we have that d + k ( p − d > d ′ and thus Clg( { rx · · · x d } ) ⊇ Clg( { rx · · · x d + k ( p − d } ) ⊇ Clg( { rx · · · x d ′ } ) andthe claim holds. (cid:3) Lemma 6.7.
Let p and q be distinct prime numbers. Let n ∈ N , let f : Z np × Z nq → Z p × Z q be an n -ary function, and let g = q p − f . Let h ∈ R ∗ [ X ] be such that R = Z Z nq p and h = g . Let h ′ be a monomial of h with coefficient r and d = tD( h ′ ) .Then it follows that: (1) if d = 0 , then r ∈ Clg( { f } ) ; (2) if d > , then rx · · · x d ∈ Clg( { f } ) .Proof. let n , h , and let f be as in the hypothesis. By Lemma 6.3, we have thatif d = 0, then r ∈ h h ′ i R and if d >
0, then rx · · · x d ∈ h h ′ i R . By Lemma 6.4, h ′ ∈ h h i R and thus, by Lemma 6.5, if d = 0, then r ∈ Clg( { f } ) holds, and if d >
0, then rx · · · x d ∈ Clg( { f } ). (cid:3) We are now ready to prove the main result of this section which allows us toprovide a bound for the lattice of all clones containing Clo( h Z pq , + i ). Proof of Theorem 1.4.
Let p and q be distinct prime numbers. Then for all 1 ≤ i ≤ p and for all 1 ≤ j ≤ q , we define ρ i : Clo L ( h Z pq , + i ) → L ( Z p , Z q ) and ψ j : Clo L ( h Z pq , + i ) → L ( Z q , Z p ) by: ρ i ( C ) := { f | there exists n ∈ N s. t. f : Z nq → Z p , f x · · · x i ∈ C } ψ j ( C ) := { f | there exists n ∈ N s. t. f : Z np → Z q , f x · · · x i ∈ C } , (6.8)for all C ∈ Clo L ( h Z pq , + i ). Furthermore, we define ρ : Clo L ( h Z pq , + i ) →L ( Z p , Z q ) and ψ : Clo L ( h Z pq , + i ) → L ( Z q , Z p ) by: ρ ( C ) := { f | there exists n ∈ N s. t. f : Z nq → Z p , f ∈ C } ψ ( C ) := { f | there exists n ∈ N s. t. f : Z np → Z q , f ∈ C } . (6.9)Let ρ : L ( h Z pq , + i ) → L ( Z p , Z q ) p +1 ×L ( Z q , Z p ) q +1 be defined by ρ ( C ) = ( ρ ( C ) , . . . ,ρ p ( C ) , ψ ( C ) , . . . , ψ q ( C )) for all C ∈ Clo L ( h Z pq , + i ). Next we prove that for all0 ≤ i ≤ p and for all 0 ≤ j ≤ q ρ i and ψ j and thus ρ are well-defined.Let C ∈ Clo L ( h Z pq , + i ). Then we have to prove that ρ i ( C ) is a ( Z p , Z q )-linearly closed clonoid. To do so let n ∈ N , f, g ∈ ρ i ( C ) [ n ] and a, b ∈ Z p . Then f x · · · x i , gx · · · x i ∈ C . From the closure with + we have that ( af + bg ) x · · · x i ∈ C and thus af + bg ∈ ρ i ( C ) [ n ] and item (1) of Definition 1.1 holds. Furthermore,let m, n ∈ N , f ∈ ρ i ( C ) [ m ] , A ∈ Z m × nq and let g : Z np → Z q be defined by: g : ( y , . . . , y n ) f ( A · ( y , . . . , y n ) t ).Then, by definition of ρ i , we have that f x · · · x i ∈ C [ s ] , where s = max ( { i, m } ).Let k = max ( { i, n } ) and let h : Z kp × Z kq → Z kp × Z kq be a linear mappingsuch that ( x , . . . , x k , y , . . . , y k ) ( x , . . . , x k , A · ( y , . . . , y n ) t , Z q , . . . , Z q ). Bydefinition of induced function we have that gx · · · x i = f x · · · x i ◦ h and hence gx · · · x i ∈ Clg( { f x · · · x i } ) as composition of f x · · · x i and linear mappings.Thus g ∈ ρ i ( C ) which concludes the proof of item (2) of Definition 1.1. Inthe same way we can prove that ρ is well-defined and ψ j is well-defined for all0 ≤ j ≤ q . Hence ρ is well-defined.Now we prove that ρ is injective. Let C, D ∈ Clo L ( h Z pq , + i ) with ρ ( C ) = ρ ( D ).Let f ∈ C [ n ] be such that f satisfies for all ( x , y ) ∈ Z np × Z nq : f (( x , y )) = ( P m ∈ [ p − n f m x m , P h ∈ [ q − n f h y h ),where { f m } [ p − n and { f h } [ q − n are sequences of functions respectively from Z nq to Z p and from Z np to Z q . Let p ∈ R ∗ [ X ] be such that p n ] = q p − f , where R = Z Z nq p . LONES ON Z pq Let h = f m x m be a monomial of p with f m = 0, let s = tD( h ). Let d ∈ N be such that if s = 0 ,
1, then 2 ≤ d ≤ p and d = s modulo p −
1. If s = 0 ,
1, then s = d . We prove that h ∈ D by case distinction.Case s = 0 ,
1: from Lemma 6.7 it follows that h ∈ C . By Definition (6.9), f m ∈ ρ s ( C ) = ρ s ( D ) and thus h ∈ D .Case s >
1: by Lemma 6.7, C ⊇ Clg( { f m x · · · x s } ). Thus, by Lemma 6.6, C ⊇ Clg( { f m x · · · x d } ) and thus f m ∈ ρ d ( C ) = ρ d ( D ). Hence f m x · · · x d ∈ D and, by Lemma 6.6, it follows that f m x m ∈ D . This yields for a generic inducedmonomial in q p − f and thus the function q p − f ∈ D . With the same strategy wecan prove that p q − f ∈ D and thus f = p q − f + q p − f ∈ D . Hence C ⊆ D . Withthe same proof we have the other inclusion and thus ρ is injective. (cid:3) Corollary 6.8.
Let p and q be distinct prime numbers. Let Q ni =1 p k i i and Q si =1 r d i i be the factorizations of respectively g = x q − − in Z p [ x ] and g = x p − − in Z q [ x ] into their irreducible divisors. Then the number k of clones containing Clo( h Z pq , + i ) is bounded by: (6.10) k ≤ p + q +2 n Y i =1 ( k i + 1) p +1 s Y i =1 ( d i + 1) q +1 . Proof.
The proof follows from the injective function of Theorem 1.4 and [Fio19,Theorem 1 . (cid:3) We can see that in the worst case this bound is equal to 2 qp + q + p , when g = x q − − Z p [ x ] and g = x p − − Z q [ x ] can both be factorized with linearfactors. There are many examples when this happens. Furthermore, a lowerbound is given by the embedding of Theorem 1.2. Proof of Corollary 1.5 .
The proof follows from Corollary 6.8 and Theorems 1.2,[Fio19, 1 . (cid:3) With these two corollaries we have found a bound for the number of clonescontaining Clo( h Z pq , + i ). With the next two results we can also find a concretebound for the arity of the generators that we need to characterized these clones. Theorem 6.9.
Let p and q be distinct prime numbers. Then a clone C containing Clo( h Z pq , + i ) is generated by the sets of functions: L := S pi =1 { rx · · · x i | r : Z q → Z p , rx · · · x i ∈ C } ∪ { r | r : Z q → Z p , r ∈ C } R := S qi =1 { ry · · · y i | r : Z p → Z q , ry · · · y i ∈ C } ∪ { r | r : Z p → Z q , r ∈ C } .Proof. Let C be a clone containing Clo( h Z pq , + i ) and let f ∈ C [ n ] be such that f satisfies for all ( x , y ) ∈ Z np × Z nq : f (( x , y )) = ( P m ∈ [ p − n f m x m , P h ∈ [ q − n f h y h ),where { f m } [ p − n and { f h } [ q − n are sequences of functions respectively from Z nq to Z p and from Z np to Z q . Let p ∈ R ∗ [ X ] be such that p n ] = q p − f , where R = Z Z nq p .Let h = f m x m be a monomial of p and let s = tD ( h ). Then, by Lemmata 6.4and 6.5, we have that h ∈ C . Furthermore, let d ∈ N be such that if s = 0 , ≤ d ≤ p and d = s modulo p −
1. If s = 0 , s = d . Then, byLemmata 6.6 and 6.7 it follows that if s = 0 Clg( h ) = Clg( { f m } ) and Clg( h ) =Clg( { f m x · · · x s } ) = Clg( { f m x · · · x d } ) otherwise. Then let us consider the( Z p , Z q )-linearly closed clonoid generated by f m . By Theorem [Fio19, Theorem1 . f : Z q → Z p such that Cig( { f } ) = Cig( { f m } ).Hence, by the embedding of Theorem 1.4, we have that Clg( { f m } ) = Clg( { f } )and Clg( { f m x · · · x i } ) = Clg( { f x · · · x i } ) for all i ∈ [ p ]. Hence h ∈ Clg( L ) andthus q p − f ∈ Clg( L ) since Clg( L ) contains every induced monomial in q p − f .In the same way we can observe that p q − f ∈ Clg( R ) and thus f = q p − f + p q − f ∈ Clg( L ) ∪ Clg( R ) and the claim holds. (cid:3) The proof of Corollary 1.6 follows directly from Theorem 6.9 and gives animportant connection between a clone C containing Clo( h Z pq , + i ) and its subsetsof generators L and R , when p and q are distinct primes. Indeed, Theorem 6.9provides a complete list of generators for a clone containing Clo( h Z pq , + i ) that isoften redundant but explains how deep is the link between clonoids and clonesin this case. We can observe that the generators in Theorem 6.9 are formed bya product of a unary function generating a ( Z p , Z q )-linearly closed clonoid anda monomial generating a clone on Z p . This justifies the use of polynomials torepresent functions of a clone containing Clo( h Z pq , + i ) which gives a differentprospective to these functions.7. Clones containing
Clo( h Z pq , + i ) which preserve π and [ π , π ] = 0In this section our aim is to characterize clones containing Clo( h Z pq , + i ) whichpreserve π and [ π , π ] = 0. We will show there is an injective function from thelattice of all clones containing Clo( h Z pq , + i ) which preserve π and [ π , π ] ≤ h Z p , + i ) and the squareof the lattice of all ( Z q , Z p )-linearly closed clonoids.Let us start showing a general expression of a function in a clone containingClo( h Z pq , + i ) which preserves π and [ π , π ] = 0. Lemma 7.1.
Let p and q be distinct prime numbers and let f : Z np × Z nq → Z p × Z q be an n -ary function which preserves π . Then f preserves [ π , π ] = 0 if andonly if there exist f c : Z np → Z q , { a m } m ∈ [ p − n ⊆ Z p and f : Z np → Z nq such thatfor all ( x , y ) ∈ Z np × Z nq , f satisfies: LONES ON Z pq (7.1) f ( x , y ) = ( X m ∈ [ p − n a m x m , h f ( x ) , y i + f c ( x )) . Proof.
Let us prove ⇒ . Let f : Z np × Z nq → Z p × Z q be an n -ary function whichpreserves π and [ π , π ] = 0. By Lemma 3.7 item (1), we have that there exist f p : Z np → Z p and f q : Z np × Z nq → Z q such that f satisfies f ( x , y ) = ( f p ( x ) , f q ( x , y ))for all ( x , y ) ∈ Z np × Z nq . Moreover, by item (3) of Lemma 3.7 we have that f isaffine in the second component. Thus let us fix a ∈ Z np . By Remark 3.4 for all y ∈ Z nq :(7.2) f q ( a , y ) = h b a , y i + c a , for some b a ∈ Z nq . Hence, using the Lagrange interpolation functions ([Fio19,Definition 4 . f q ( x , y ) = X a ∈ Z np f a ( x )( h b a , y i + c a ) , for all y ∈ Z nq . Thus f satisfies (7.1) with f = P a ∈ Z np f a b a and f c = P a ∈ Z np f a c a .Moreover, by Remark 3.3, there exists a sequence { a m } m ∈ Z np such that f ( x ) = P m ∈ [ p − n a m x m for all x ∈ Z np . Hence the ⇒ implication holds. For the ⇐ implication suppose that there exist f c : Z np → Z q , { a m } m ⊆ [ p − n ⊆ Z p and f : Z np → Z nq such that for all ( x , y ) ∈ Z np × Z nq , f satisfies (7.1). Let x , x , x , x ∈ Z np and y , y , y , y ∈ Z nq such that x = x = x = x and y − y + y = y . Then: f ( x , y ) − f ( x , y ) + f ( x , y ) = ( X m ∈ [ p − n a m x m , h f ( x ) , y − y + y i + f c ( x ))= f ( x , y )(7.4)and thus f preserves [ π , π ] = 0. (cid:3) Let us now provide a proof of Theorem 1.7. Let us consider the functions ρ : Clo ⋄ ( h Z pq , + i ) → Clo L ( h Z p , + i ) as defined in the proof of Theorem 1.3, ψ , ψ : Clo L ( h Z pq , + i ) → L ( Z p , Z q ) as defined in the proof of Theorem 1.4. Proof of Theorem 1.7.
Let p and q be distinct prime numbers and let Clo ′ ( h Z pq , + i )be the lattice of all clones containing Clo( h Z pq , + i ) which preserve π and [ π , π ] ≤
0. Let ψ : Clo ′ ( h Z pq , + i ) → Clo L ( h Z p , + i ) × L ( Z p , Z q ) be defined by ψ ( C ) :=( ρ ( C ) , ψ ( C ) , ψ ( C )) for all C ∈ Clo ′ ( h Z pq , + i ). With the same proof of Theo-rems 1.3 and 1.7 we see that ρ , ψ , ψ and thus ψ are well-defined. Next we prove that ψ is injective. To this end let C, D ∈ Clo ′ ( h Z pq , + i ) besuch that ψ ( C ) = ψ ( D ). Let us suppose that f ∈ C [ n ] . By Lemma 7.1, thereexist f c : Z np → Z q , { a m } m ∈ [ p − n ⊆ Z p and f : Z np → Z nq such that for all( x , y ) ∈ Z np × Z nq , f satisfies:(7.5) f ( x , y ) = ( X m ∈ [ p − n a m x m , h f ( x ) , y i + f c ( x )) . It is clear that the functions { f i : Z np × Z nq → Z p × Z q } ≤ i ≤ n , f ′ c : Z np × Z nq → Z p × Z q ,and t : Z np × Z nq → Z p × Z q such that for all ( x , y ) ∈ Z np × Z nq , f i ( x , y ) =(0 , ( f ( x )) i ( y ) i ), f ′ c ( x , y ) = (0 , f c ( x )), and t ( x , y ) = ( P m ∈ [ p − n a m x m ,
0) are in C . Thus π Z p × Z q p ◦ t | Z np × { } ∈ ρ ( C ) = ρ ( D ), ( f ) i ∈ ψ ( C ) = ψ ( D ) for all i ∈ [ n ], and f c ∈ ψ ( C ) = ψ ( D ). Hence, by definition of ρ, ψ , ψ , this impliesthat p, f , . . . , f n , f ′ c ∈ D . Hence f = t + P ni =1 f i + f ′ c ∈ D . Thus C ⊆ D andin the same way we can prove the other inequality. Hence C = D and the claimholds. (cid:3) Corollary 7.2.
Let p and q be distinct prime numbers. Let Q ni =1 p k i i be the factor-ization of the polynomial g = x p − − in Z q [ x ] into its irreducible divisors. Thenthe number of clones containing Clo( h Z pq , + i ) which preserve π and [ π , π ] ≤ is bounded from above by ( n ( p ) + 3)(2 Q ni =1 ( k i + 1)) , where n ( k ) is the numberof divisors of k − .Proof. We know from [Kre19, Corollary 1 .
3] and [Pos41] that number of clonescontaining Clo( h Z p , + i ) is n ( p ) + 3 where n ( p ) is the number of divisors of p − .
3] that the cardinality of thelattice of all ( Z q , Z p )-linearly closed clonoids is 2 Q ni =1 ( k i + 1), where { k i } i ∈ [ n ] arethe exponents of the factorization of the polynomial g = x p − − Z q [ x ]. Thenthe bound directly follows from the injective function of Theorem 1.7. (cid:3) We can observe that the bound of Corollary 7.2 is not reached. In particularthe images of ψ and ψ , are ( Z q , Z p )-linearly closed clonoids that satisfy somefurther closure properties. Indeed, both are closed from the right not only undercomposition with linear mappings but also with functions of the clone image of ρ .Furthermore, the image of ψ contains constants and is closed under point-wiseproduct. The image of ψ is closed under point-wise product with functions in C . These structures are related to vector subspaces of Z qp that are closed underHadamard product. We will not go into details in the case, nevertheless we thinkthat they are interesting structures.In general we cannot conclude that ρ is an embedding since the join of two( Z q , Z p )-linearly closed clonoids that satisfy the previous properties could alsonot satisfy them. LONES ON Z pq Acknowledgements
The author thanks Erhard Aichinger, who inspired this paper, and SebastianKreinecker for many hours of fruitful discussions.
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Stefano Fioravanti, Institut f¨ur Algebra, Johannes Kepler Universit¨at Linz,4040 Linz, Austria
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