Clones of Compatible Operations on Rings Z_{p^k}}
CClones of Compatible Operations on Rings Z p k M IROSLAV P LO ˇ S ˇ CICA (cid:63) , I VANA V ARGA † Institute of Mathematics, ˇSaf´arik’s University, Koˇsice, Slovakia Institute of Mathematics, ˇSaf´arik’s University, Koˇsice, Slovakia
Received 24 June 2020; awaiting publication
We investigate the lattice I ( n ) of clones on the ring Z n betweenthe clone of polynomial functions and the clone of congruencepreserving functions. The crucial case is when n is a primepower. For a prime p , the lattice I ( p ) is trivial and I ( p ) is knownto be a -element lattice. We provide a description of I ( p ) . Toachieve this result, we prove a reduction theorem, which says that I ( p k ) is isomorphic to a certain interval in the lattice of cloneson Z p k − . Key words: congruence, clone, polynomial A clone on a set A is a family of operations, which contains all projectionsand is closed under composition. The family of all clones on A forms a lattice.An n -ary operation on an algebra A is called compatible or congruencepreserving if for x , . . . , x n , y . . . , y n ∈ A , ( x , y ) , . . . , ( x n , y n ) ∈ θ im-plies ( f ( x , . . . , x n ) , f ( y , . . . , y n )) ∈ θ for every θ ∈ Con A . It is clearthat all compatible operations form a clone, denoted by Comp( A ) . Thisclone includes the clone P( A ) of all polynomial operations on A . Hence, P( A ) ⊆ Comp( A ) . If the equality P( A ) = Comp( A ) holds, then the alge-bra A is called affine complete . A lot of research has been devoted to affine (cid:63) email: [email protected] † email: [email protected] a r X i v : . [ m a t h . R A ] J u l ompleteness for various types of algebras. Some survey can be find in themonograph [8] of Kaarli and Pixley.The algebras considered in this paper are the rings Z n of integers modulo n . These rings are known to be affine complete if and only if n is squarefree (aproduct of distinct primes). This can be deduced from general theorems aboutaffine complete rings, see [8]. For n that is not squarefree, we have P( Z n ) (cid:40) Comp( Z n ) , and a natural question is to describe the interval between P( Z n ) and Comp( Z n ) . Clearly, this interval is a lattice, let us denote it by I ( n ) . Thecrucial case is when n = p k , a power of a prime p , k ≥ . (See Theorem2.4.) Problem 1.1
Describe the lattice I ( p k ) for a prime p and k ≥ . The answer to this problem seems not to be known for k > . The case k = 2 has been solved by Remizov in [13], who showed that I ( p ) is a -element lattice. Alternative proofs of this result can be found in Bulatov [4],and also in our present paper. (See Theorem 4.1.) For the case k > , onlypartial results are available, see [13], [5], [6], and [10]. Especially, it is knownthat the lattice I ( p k ) for k > is infinite.In the present paper we prove a reduction theorem, which says that thelattice I ( p k ) is isomorphic to the interval between the clones E ( Z p k − ) and Comp( Z p k − ) , where E ( Z p k − ) is the clone on Z p k − generated by addi-tion, constants and the binary operation g ( x, y ) = pxy . Notice that the clone E ( Z p k − ) is smaller than P( Z p k − ) , but includes all polynomials of the ad-ditive group Z p k − . So, the description of I ( p k ) depends on a description ofclones on Z p k − containing all group polynomials. For k = 3 such cloneshave been completely described by Bulatov in [4]. (See also Meshchaninov’spaper [11].) Relying on this paper we are able to provide a complete descrip-tion of the lattice I ( p ) .Research in the clone theory connected with the modular arithmetics hasa long and rich history. Clones of polynomials on Z n were studied in papers[15], [16], [17], [2], [3] of Salomaa, Szendrei, Bulatov, and others. From amore recent research relevant to our topic, we would like to mention the re-sults of Idziak ([7]), Aichinger and Mayr ([1]), and Mayr ([9]). These papersinvestigate clones containing a group operation on a given finite set.The clone of all compatible functions has been studied also for other kindsof algebras. For instance, the paper [12] describes generators of this clone fordistributive lattices.Elements of Z n will be denoted , , . . . , n − . Congruences on the ring Z n are the usual congruences modulo d for every d | n . For vectors in Z kn we2dopt the convention that x = ( x , . . . , x k ) , l = ( l , . . . , l k ) , etc. Z mn Let m, n be integers such that gcd( m, n ) = 1 . It is well known that the ring Z mn is isomorphic to the product of rings Z m × Z n . Congruences of the ring Z m × Z n are θ q,r such that (( x, y ) , ( u, v )) ∈ θ q,r iff x ≡ u (mod q ) and y ≡ v (mod r ) , where q is a divisor of m and r is a divisor of n . Let I mn be the interval ofclones between P( Z m × Z n ) and Comp( Z m × Z n ) . Clearly, the lattice I mn is isomorphic to I ( mn ) and we show that it is isomorphic to I ( m ) × I ( n ) . Lemma 2.1
Let m, n be coprime numbers. Compatible operations on thering Z m × Z n are precisely operations of the form f (( x , y ) , . . . , ( x l , y l )) = ( f ( m ) ( x ) , f ( n ) ( y )) , where f ( m ) is compatible on Z m and f ( n ) is compatible on Z n . P r o o f. Let f be an l -ary compatible operation on Z m × Z n . Let p denotethe projection from Z m × Z n to Z m . Then θ = Ker( p ) is a congruenceon Z m × Z n . Let p denote the projection to Z n , then θ = Ker( p ) is acongruence on Z m × Z n . Since f preserves the congruences θ and θ ,wehave f (( x , y ) , . . . , ( x l , y l )) θ f (( x , , . . . , ( x l , f (( x , y ) , . . . , ( x l , y l )) θ f ((0 , y ) , . . . , (0 , y l )) for arbitrary x ∈ ( Z m ) l , y ∈ ( Z n ) l . Set f ( m ) ( x ) = p ( f (( x , , . . . , ( x l , f ( n ) ( y ) = p ( f ((0 , y ) , . . . , (0 , y l ))) . Then p ( f (( x , y ) , . . . , ( x l , y l ))) = f ( m ) ( x ) p ( f (( x , y ) , . . . , ( x l , y l ))) = f ( n ) ( y ) . We proved that there exist operations f ( m ) , f ( n ) , such that f = ( f ( m ) , f ( n ) ) .It is clear that f preserves θ q,r if and only if f ( m ) preserves the congruencemodulo q and f ( n ) preserves the congruence modulo r . So, f is compatibleif and only if f ( m ) , f ( n ) are compatible.It is easy to see the following. 3 emma 2.2 For every l -ary operation f and all k -ary operations g , . . . , g l from a clone C ∈ I mn , the following holds: ( f ( g , . . . , g l )) ( m ) = f ( m ) ( g ( m )1 , . . . , g ( m ) l ) . Lemma 2.3
Let m, n be coprime numbers.(i) Let C ∈ I mn . Then the set C m = { f ( m ) | f ∈ C } ∈ I ( m ) and the set C n = { f ( n ) | f ∈ C } ∈ I ( n ) .(ii) Let D ∈ I ( m ) and E ∈ I ( n ) . Then D × E = { f | f ( m ) ∈ D, f ( n ) ∈ E } ∈ I mn . (iii) For every C ∈ I mn : C = C m × C n .(iv) For every D ∈ I ( m ) , E ∈ I ( n ) : ( D × E ) m = D, ( D × E ) n = E . P r o o f. (i) It follows from Lemma 2.1 that the set C m is a subset of Comp( Z m ) . It is closed under composition (from Lemma 2.2) and containsall projections, addition, multiplication and constants, because C containsprojections, addition, multiplication and constants. Analogously we can showthat C n ∈ I ( n ) .(ii) It follows from Lemma 2.1 that the set D × E is a subset of the clone Comp( Z m × Z n ) . It is closed under composition (from Lemma 2.2) andcontains all projections. Let g ( m ) ( x , x ) = x + x and g ( n ) ( y , y ) = y + y . Then we have g (( x , y ) , ( x , y )) = ( g ( m ) ( x , x ) , g ( n ) ( y , y )) =( x + x , y + y ) , therefore D × E contains addition. Similarly, D × E contains multiplication and constants.(iii) We want to prove that C = { f | f ( m ) ∈ C m , f ( n ) ∈ C n } . If f ∈ C ,then f ( m ) ∈ C m and f ( n ) ∈ C n directly from the definition. Conversely, let f ( m ) ∈ C m and f ( n ) ∈ C n . That means there exist functions g, h ∈ C , suchthat g ( m ) = f ( m ) and h ( n ) = f ( n ) . Using the Chinese remainder theorem,there exist natural numbers a , b such that a ≡ m ) , a ≡ n ) , b ≡ n ) , and b ≡ m ) . Put f = g + g + . . . + g (cid:124) (cid:123)(cid:122) (cid:125) a-times + h + h + . . . + h (cid:124) (cid:123)(cid:122) (cid:125) b-timesSince C contains the addition, we have f ∈ C . Moreover, f ( m )1 ( x ) = g ( m ) ( x ) · a + h ( m ) ( x ) · b = g ( m ) ( x ) = f ( m ) ( x ) f ( n )1 ( x ) = g ( n ) ( x ) · a + h ( n ) ( x ) · b = h ( n ) ( x ) = f ( n ) ( x ) , hence f = f ∈ C .(iv) Let f ∈ D × E , then f ( m ) ∈ D from the definition. Let g ∈ D , thenthere exists an operation f ∈ D × E , such that f ( m ) = g . So, g ∈ ( D × E ) m ,which proves the statement. Analogously we can show that ( D × E ) n = E .Lemma 2.3 shows that the assignments C (cid:55)→ ( C m , C n ) and ( D, E ) (cid:55)→ D × E are mutually inverse bijections between I mn and I ( m ) × I ( n ) . Clearly,they are order preserving, so the lattices I mn and I ( m ) × I ( n ) are isomorphic.Using induction, we obtain the following result. Theorem 2.4
Let n = p α · . . . · p α m m , where p , . . . , p m are distinct primes.Then the lattice I ( n ) is isomorphic to the product of the lattices I ( p α i i ) , i =1 , . . . , m . The above theorem can be also deduced from [13], in a different formal-ism. We include the proof here for the sake of completeness.So, in order to describe I ( n ) it suffices to investigate the lattices I ( p k ) fora prime p and k ≥ . Z p k TO Z p k − Let k ≥ be a fixed integer. Let M = { lp | l ∈ { , . . . , p k − − }} denotethe set of multiples of p in Z p k . Lemma 3.1
There exists a polynomial function G on the ring Z p k satisfying G ( x ) = (cid:26) , if x ∈ M n , otherwise. P r o o f. For every x ∈ M , we have (cid:89) c ∈ Z pk \ M ( x − c ) = (cid:89) c ∈ Z pk \ M c = α ,where the latter is invertible and independent of x . It is not difficult to provethat α is equal to ( − for p > and for p = 2 . Then the polynomial G ( x ) = α − n n (cid:89) i =1 (cid:89) c ∈ Z pk \ M ( x i − c ) L = { , , . . . , p − } ⊆ Z p k . For every x ∈ Z np k there exists a unique c ∈ L n such that x − c ∈ M n . Thus, Lemma 3.1 enables the following easydecomposition. Lemma 3.2
For every n -ary function f on Z p k , f ( x ) = (cid:88) c ∈ L n f ( x ) G ( x − c ) . Let P( M ) denote the clone on M generated by addition and multiplicationmodulo p k and constant operations. Let Comp( M ) denote the clone thatconsists of all operations that preserve congruences modulo p , . . . , p k − . Itis obvious that P( M ) ⊆ Comp( M ) .Now we are going to prove that the interval in the lattice of clones between P( M ) and Comp( M ) is isomorphic to I ( p k ) .We say that f preserves M , if f ( x ) ∈ M whenever x ∈ M n . For anyclone C ∈ I ( p k ) (that is, for every clone between P( Z p k ) and Comp( Z p k ) )we define C M = { f (cid:22) M | f ∈ C, f preserves M } . We show that assignment C (cid:55)→ C M is the required isomorphism. Lemma 3.3
For every clone C ∈ I ( p k ) , the set C M is a clone between P( M ) and Comp( M ) . P r o o f. It is clear that C M is a clone. The rest follows from the fact,that the restriction of every f ∈ Comp( Z p k ) belongs to Comp( M ) , and theaddition and multiplication on Z p k restrict to the addition and multiplicationon M .Conversely, for every clone K on M between P( M ) and Comp( M ) wedefine C ( K ) = { f ∈ Comp( Z p k ) | ∀ a ∈ Z np k : ( f ( x + a ) − f ( a )) (cid:22) M ∈ K } . Notice that the compatible operation f preserves the congruence mod p ,which implies that the operation f ( x + a ) − f ( a ) preserves M . We obtainthat the restriction ( f ( x + a ) − f ( a )) (cid:22) M is in Comp( M ) . Lemma 3.4
For every clone K between P( M ) and Comp( M ) , the set C ( K ) is a clone in I ( p k ) . n -ary projection f ( x ) = x i on Z p k and every a ∈ Z np k ,the operation f ( x + a ) − f ( a ) = x i + a i − a i = x i is a projection on M andhence belongs to K . Therefore, C ( K ) contains all projections.2. To show that C ( K ) is closed under composition, consider operations f , g , . . . , g n ∈ C ( K ) , with f n -ary and all g i m -ary. Let a ∈ Z mp k . Then ( g ( a ) , . . . , g n ( a )) ∈ Z np k . Since f ∈ C ( K ) , the operation h ( x ) = ( f ( x + ( g ( a ) , . . . , g n ( a )) − f ( g ( a ) , . . . , g n ( a ))) (cid:22) M belongs to K . Further, for every i , the operation h i ( x ) = ( g i ( x + a ) − g i ( a )) (cid:22) M belongs to K because g i ∈ C ( K ) . Since K is closed under composition, wehave h ( h , . . . , h n )( x ) ∈ K . For every x ∈ M m we have h ( h , . . . , h n )( x ) = h ( h ( x ) , . . . , h n ( x )) = h ( g ( x + a ) − g ( a ) , . . . , g n ( x + a ) − g n ( a )) = f ( g ( x + a ) , . . . , g n ( x + a )) − f ( g ( a ) , . . . , g n ( a )) Therefore f ( g , . . . , g n )( x ) belongs to C ( K ) .3. It is easy to check that C ( K ) contains the addition.4. Let f ( x, y ) = x · y , a , a ∈ Z p k . Then f ( x + a , y + a ) − f ( a , a ) =( x + a ) · ( y + a ) − ( a · a ) = x · y + x · a + y · a . The restriction ofthis function to M belongs to K , because it is a polynomial. (Notice that x · a can be replaced by the sum of a copies of x , and similarly for y · a .)Therefore f ∈ C ( K ) .5. Every constant operation f ( x ) = c belongs to the clone C ( K ) because f ( x + a ) − f ( a ) = 0 , and the restriction of this function belongs to K .6. The inclusion C ( K ) ⊆ Comp( Z p k ) follows directly from the defini-tion. Lemma 3.5 K = C ( K ) M for every clone K between the clones P( M ) and Comp( M ) . P r o o f. Let h ∈ K be an n -ary operation on M . Define f ( x ) = (cid:26) h ( x ) , if x ∈ M n , otherwise.7hen f preserves M and h = f (cid:22) M . We claim that f ∈ C ( K ) . Theoperation f is compatible because h is compatible. We need to show that ( f ( x + a ) − f ( a )) (cid:22) M ∈ K holds for every a ∈ Z np k . For a ∈ M n we have ( f ( x + a ) − f ( a )) (cid:22) M = h ( x + a ) − h ( a ) ∈ K and for a / ∈ M n we have ( f ( x + a ) − f ( a )) (cid:22) M = 0 ∈ K . This proves that K ⊆ C ( K ) M .Conversely, let h ∈ C ( K ) M , which means that there exists f ∈ C ( K ) ,such that h = f (cid:22) M and f preserves M . Using the definition of C ( K ) with a = we obtain that ( f ( x + ) − f ( )) (cid:22) M ∈ K . Since f ( ) ∈ M , and K contains constants and the addition, we have h ( x ) = f ( x ) (cid:22) M ∈ K . Thisproves that C ( K ) M ⊆ K . Lemma 3.6
Let
C, D ∈ I ( p k ) . Then C ⊆ D if and only if C M ⊆ D M . P r o o f. If C ⊆ D then it is obvious that also C M ⊆ D M . Conversely,suppose that C M ⊆ D M . Let f ∈ C be an n -ary operation. Then, forevery c ∈ { , . . . , p − } n , the operation f ( x + c ) belongs to C . The n -aryconstant function f ( c ) is also in C and therefore f ( x + c ) − f ( c ) ∈ C . Thecompatibility of f implies that this function preserves M , so the operation ( f ( x + c ) − f ( c )) (cid:22) M belongs to C M ⊆ D M . Consequently, there existsan operation g c ∈ D such that g c ( x ) = f ( x + c ) − f ( c ) for every x ∈ M n .Using the polynomial G from Lemma 3.1 we get g c ( x ) G ( x ) = ( f ( x + c ) − f ( c )) G ( x ) , which holds for every c and every x ∈ Z np k . (It is trivial for x / ∈ M n .) Afterthe substitution x = u − c we obtain g c ( u − c ) G ( u − c ) = ( f ( u ) − f ( c )) G ( u − c ) for every u ∈ Z np k . Now we use Lemma 3.2: f ( u ) = (cid:88) c f ( u ) G ( u − c ) = (cid:88) c g c ( u − c ) G ( u − c ) + (cid:88) c f ( c ) G ( u − c ) and from this expression it follows that f ∈ D . Theorem 3.7
The lattice I ( p k ) is isomorphic to the interval between P( M ) and Comp( M ) . C (cid:55)→ C M is an order embedding.By Lemma 3.5, it is also surjective.So, the assignment C (cid:55)→ C M is a bijection and hence has a unique inverse.According to Lemma 3.5, this inverse is the assignment K (cid:55)→ C ( K ) . Hence,we also have the following assertion. Lemma 3.8 D = C ( D M ) for every D ∈ I ( p k ) . Lemma 3.9
Let K be a clone on M generated by addition, multiplication,constants and operations { h i | i ∈ I } . Then C ( K ) is generated by addition,multiplication, constants and operations f i ( x ) = (cid:26) h i ( x ) , if x ∈ M n , otherwise. P r o o f. Let D be generated by such operations on Z p k . These generatorsbelong to C ( K ) , therefore D ⊆ C ( K ) . It is clear, that D M ⊇ K because D M contains all generators of K and that yields D = C ( D M ) ⊇ C ( K ) .As the second step in our reduction from Z p k to Z p k − we now show thatthe interval between P( M ) and Comp( M ) is isomorphic to a certain intervalin the lattice of clones on Z p k − . The key is in the following construction.Let f : ( Z p k − ) n → Z p k − be an n -ary operation and define the operation f ∗ : M n → M for every l i ∈ Z p k − as follows f ∗ ( l p, . . . , l n p ) = f ( l , . . . , l n ) · p. Recall that we identify Z p k − as the set { , , . . . , p k − − } . The numbers l i on the left hand side of the above equation are treated as elements of Z p k . Thisdefinition is correct, as every element of M is equal to lp for some l ∈ Z p k − .It is easy to see the following assertion. Lemma 3.10
An operation f : ( Z p k − ) n → Z p k − belongs to Comp( Z p k − ) if and only if f ∗ ∈ Comp( M ) . Lemma 3.11
Let m, n ∈ N . For every n -ary operation f and all m -aryoperations g , . . . , g n on Z p k − , the following holds: ( f ( g , . . . , g n )) ∗ = f ∗ ( g ∗ , . . . , g ∗ n ) . P r o o f. For every l = ( l , . . . , l m ) ∈ ( Z p k − ) m we compute ( f ( g , . . . , g n )) ∗ ( l p ) = f ( g , . . . , g n )( l ) · p = f ( g ( l ) , . . . , g n ( l )) · p, f ∗ ( g ∗ ( l p ) , . . . , g ∗ n ( l p )) = f ∗ ( g ( l ) · p, . . . , g n ( l ) · p ) = f ( g ( l ) , . . . , g n ( l )) · p. We have the required equality.Let E ( Z p k − ) denote the clone on the ring Z p k − generated by constants,addition and the binary operation pxy . We show that the interval between P( M ) and Comp( M ) and the interval between E ( Z p k − ) and Comp( Z p k − ) are isomorphic.Let C be a clone between E ( Z p k − ) and Comp( Z p k − ) and let C ∗ denotethe set C ∗ = { f ∗ | f ∈ C } . Lemma 3.12
For every clone C between E ( Z p k − ) and Comp( Z p k − ) , C ∗ is a clone between P( M ) and Comp( M ) . P r o o f. 1. If f is a projection on Z p k − , then f ∗ is the same projection on M . By Lemma 3.11, C ∗ is closed under composition. Thus, C ∗ is indeed aclone.2. If f is the addition on Z p k , then f ∗ is the addition mod p k on M . If g ( x, y ) = pxy on Z p k − , then g ∗ ( xp, yp ) = g ( x, y ) · p = ( pxy ) p = px · py is the multiplication modulo p k on M . Further, every constant operation h ( x ) = c ∈ Z p k − belongs to C , so h ∗ ( lp ) = cp is a constant operationon M . We have obtained that C ∗ ⊇ P( M ) .3. Every f ∈ C preserves congruences modulo p, . . . , p k − . Conse-quently, then f ∗ preserves congruences modulo p , . . . , p k − . The congru-ence mod p is trivial on M . Hence, C ∗ ⊆ Comp( M ) .Clearly, f ∗ = g ∗ if and only if f = g . Hence, the assignment C (cid:55)→ C ∗ isan order embedding. Now we show its surjectivity. Lemma 3.13
For any clone D between P( M ) and Comp( M ) , the set C = { h ∈ Comp( Z p k − ) | h ∗ ∈ D } is a clone between E ( Z p k − ) and Comp( Z p k − ) . Moreover, D = C ∗ . P r o o f. It is clear that C is a clone. (The closedness under compositionfollows from Lemma 3.11.) Moreover, if f is the addition on Z p k − , then f ∗ is the addition on M , which belongs to D , so f ∈ C . Similarly, if g ( x, y ) = xy on Z p k − , then g ∗ is the multiplication on M , so g ∗ ∈ D and hence g ∈ C . If h is a constant operation on Z p k − , then h ∗ is a constant operation on M and, again, h ∗ ∈ D implies h ∈ C . We have proved that E ( Z p k − ) ⊆ C .By Lemma 3.10 we have C ⊆ Comp( Z p k − ) .It remains to prove that D = C ∗ . The inclusion C ∗ ⊆ D is trivial. Con-versely, let f ∈ D . Then there is an operation h on Z p k − such that f ( x p ) = h ( x ) · p . Clearly, f = h ∗ . Lemma 3.10 implies that h ∈ Comp( Z p k − ) , so h ∈ C and f ∈ C ∗ .As a consequence of previous lemmas we state the following theorem. Theorem 3.14
The lattice I ( p k ) is isomorphic to the interval between clones E ( Z p k − ) and Comp( Z p k − ) . The isomorphism in our Theorem maps a clone K between E ( Z p k − ) and Comp( Z p k − ) first into the clone K ∗ and then, by Theorem 3.7, into C ( K ∗ ) . We also have a correspondence between generators. If { f i | i ∈ I } is a generating set of K , then (by Lemma 3.11) { f ∗ i | i ∈ I } is a generatingset for K ∗ . The generating set of C ( K ∗ ) is then described by Lemma 3.9. k = 2 AND k = 3 The interval between E ( Z p k − ) and Comp( Z p k − ) is known for k = 2 and k = 3 . We can use this knowledge to describe all clones between P ( Z p k ) and Comp( Z p k ) .If k = 2 , then the operation g ( x, y ) = pxy on Z p is trivially zero. So, E ( Z p ) is the clone of all polynomials of the group ( Z p , +) . It is well knownthat this clone is maximal, which means that it is covered by the clone of alloperations on the set Z p . (It can be deduced from the well known Rosenberg’sclassification in [14].) The clone of all operations coincides with Comp( Z p ) ,since the ring Z p has only trivial congruences and therefore all operations arecompatible. We obtain the following result. Theorem 4.1
The interval between P( Z p ) and Comp( Z p ) has only twoelements. The case k = 3 is much more complicated. The interval between E and Comp( Z p ) is only known from Bulatov’s paper [4]. (We write E instead of E ( Z p ) .) In fact, Bulatov described all clones, which contain polynomials ofthe group ( Z p , +) . The lattice of these clones is depicted below. Each clone11s determined by a set of generators, which always contains the addition andthe constants. Notice that the picture includes the fact stated in Theorem 4.1. Theorem 4.2
The lattice of clones between P ( Z p ) and Comp( Z p ) is iso-morphic to the interval E and Comp( Z p ) on the picture below. (cid:115)(cid:115)(cid:115)(cid:115)(cid:115) (cid:115)(cid:115)(cid:115) (cid:115) (cid:115)(cid:115) (cid:115) (cid:115)(cid:115) ...... (cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16) ... (cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16) P grp ( Z p , +) E E p E p +1 E N p N p +1 NF P( Z p ) F F Comp( Z p ) O ( Z p ) Now we list the generators of all clones K between E and Comp( Z p ) (taken from [4]), as well as the generators of the corresponding clones Φ( K ) = C ( K ∗ ) between P ( Z p ) and Comp( Z p ) . The generators of Φ( K ) are constructed by the process described at the end of the previous section.The definitions are as follows. The operation h on Z p is defined by theformula h ( x, y ) = (cid:26) klp, if x = kp, y = lp for some k, l ∈ { , . . . , p − } , otherwise.The j -ary operation ξ j on Z p is defined by ξ j ( x ) = (cid:26) k . . . k j p , if x = k p for some k ∈ { , . . . , p − } j , otherwise. 12otice that ξ is the restriction of the usual multiplication to M . It is a poly-nomial of the ring Z p , so Φ( E ) = P( Z p ) .Next we define operations π , ψ , ρ , ϕ and τ on Z p . π ( x ) = (cid:26) pk p , if x = kp for k ∈ { , . . . , p − } , otherwise. ψ ( x, y ) = (cid:26) pk p l p , if x = kp, y = lp for k, l ∈ { , . . . , p − } , otherwise. ρ ( x, y ) = (cid:26) pk p ( l p − l ) , if x = kp, y = lp for k, l ∈ { , . . . , p − } , otherwise. ϕ ( x, y ) = (cid:26) klp , if x = kp , y = lp for k, l ∈ { , . . . , p − } , otherwise. τ ( x, y ) = (cid:26) klp, if x = kp, y = lp for k, l ∈ { , . . . , p − } , otherwise.The generators of all clones are in the following table. We only list theadditional generators (besides addition and constants for K , besides addition,multiplication and constants for Φ( K ) ). The clones E and N are not in thetable, they are the union of all E j and N j , respectively. K generators of K generators of Φ( K ) E j px . . . x j ξ j N j px . . . x j , x p ξ j , πF x p y p ψF x p ( y p − y ) ρF h ϕ P( Z p ) xy τ Comp( Z p ) xy , h τ , ϕ This work has been supported by Slovak VEGA grant 1/0097/18.13
EFERENCES [1] E. Aichinger, P. Mayr,
Polynomial clones on groups of order pq , Acta Math. Hungar. (2007), 267-285.[2] A. A. Bulatov, Polynomial reducts of modules I. Rough classification , Multiple-valuedLogic (1998), 135-154.[3] A. A. Bulatov, Polynomial reducts of modules II. Algebras of Primitive and nilpotent func-tions , Multiple-valued Logic (1998), 173-193.[4] A. A. Bulatov, Polynomial clones containing the Mal’tsev operation of the groups Z p and Z p × Z p , Multiple-valued Logic (2002), 193-221.[5] G. P. Gavrilov, On the overstructure of the class of polynomials of multivalued logics (inRussian), Diskretnaja Matematika (1996), 90-97.[6] G. P. Gavrilov, On closed classes of multi-valued logic that contain the class of polynomials (in Russian), Diskretnaja Matematika , (1997), 12-23.[7] P. Idziak, Clones with Mal’tsev operation , Internat. J. of Algebra and Computation (1999), 213-226.[8] K. Kaarli, A. F. Pixley, Polynomial completeness in algebraic systems , Chapman & Hall/CRC, 2001.[9] P. Mayr,
Polynomial clones on squarefree groups , Internat. J. of Algebra and Computation (2008), 759-777.[10] D. G. Meshchaninov, On some properties of the overstructure of classes of polynomials in P k (in Russian), Mat. Zametki (1988), 673-681.[11] D. G. Meshchaninov, A family of Closed Classes in k -Valued Logic , Moscow UniversityComputational Mathematics and Cybernetics (2019), 25-31.[12] M. Ploˇsˇcica, M. Haviar, Congruence-preserving functions on distributive lattices , AlgebraUniversalis (2008), 179-196.[13] A. B. Remizov, On the overstructure of closed classes of polynomials modulo k (in Rus-sian), Diskretnaja Matematika (1989), 3-15.[14] I. G. Rosenberg, ¨Uber die funktionalle Vollst¨andigkeit in den mehrwertigen Logiken , Roz-pravy ˇCeskoslovensk´e Akademie vˇed, Ser. Math. Nat. Sci. (1970), 3-93.[15] A. A. Salomaa, On infinitely generated sets of operations in finite algebras , Ann. Univ.Turku, Ser. AI (1964), 1-12.[16] ´A. Szendrei, Idempotent reducts of Abelian groups,
Acta Sci. Math. (Szeged) (1976),171-182.[17] ´A. Szendrei, Clones of linear operations on finite sets , in: Finite algebra and multiple-valued logic, Colloq. Math. Soc. J. Bolyai (1977), 693-738.(1977), 693-738.