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Closed sets of finitary functions between products of finite fields of coprime order

Closed sets of finitary functions between products of finite fields of coprime order Algebra Univers. (2021) 82:61 c 2021 The Author(s) Algebra Universalis https://doi.org/10.1007/s00012-021-00748-z Closed sets of finitary functions between products of finite fields of coprime order Stefano Fioravanti Abstract. We investigate the finitary functions from a finite product of m n finite fields F = K to a finite product of finite fields F = F, q p j=1 j i=1 i where |K| and |F| are coprime. An (F, K)-linearly closed clonoid is a subset of these functions which is closed under composition from the right and from the left with linear mappings. We give a characterization of these × K × subsets of functions through the F [K ]-submodules of F ,where K is p p the multiplicative monoid of K = F . Furthermore we prove that i=1 i each of these subsets of functions is generated by a set of unary functions and we provide an upper bound for the number of distinct (F, K)-linearly closed clonoids. Mathematics Subject Classification. 08A40. Keywords. Clonoids, Clones. 1. Introduction Since P. Hall’s abstract definition of a clone the problem to describe sets of finitary functions from a set A to a set B which satisfy some closure properties has been a fruitful branch of research. E. Post’s characterization of all clones on a two-element set [12] can be considered as a foundational result in this field, which was developed further, e.g., in [13, 11, 15, 10]. Starting from [9], 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 a finite product m n of finite fields F = K to a finite product of finite fields F = F, q p j=1 j i=1 i where |K| and |F| are coprime. The sets of functions we are interested in are closed under composition from the left and from the right with linear mappings. Thus we consider sets of functions with different domains and codomains; such Presented by R. P¨ oschel. The research was supported by the Austrian Science Fund (FWF): P29931. 0123456789().: V,-vol 61 Page 2 of 12 S. Fioravanti Algebra Univers. sets are called clonoids and are investigated, e.g., in [2]. Let B be an algebra, and let A be a non-empty set. For a subset C of B and k ∈ N,welet n∈N [k] A C := C ∩ B . According to Definition 4.1of[2] we call C a clonoid with source set A and target algebra B if [k] A (1) for all k ∈ N: C is a subuniverse of B ,and k [k] (2) for all k, n ∈ N, for all (i ,...,i ) ∈{1,...,n} , and for all c ∈ C ,the 1 k n  [n] function c : A → B with c (a ,...,a ):= c(a ,...,a ) lies in C . 1 n i i 1 k By (1) every clonoid is closed under composition with operations of B on the left. In particular we are dealing with those clonoids whose target algebra is the ring F that are closed under composition with linear mappings i=1 i from the right side. m s Definition 1.1. Let m, s ∈ N and let K = K , F = F be prod- j i j=1 i=1 ucts of fields. An (F, K)-linearly closed clonoid is a non-empty subset C of m k j=1 j F with the following properties: k∈N i=1 i [n] (1) for all n ∈ N, a , b ∈ F ,and f, g ∈ C : i=1 [n] a f + bg ∈ C ; n×l [n] l (2) for all l, n ∈ N, f ∈ C ,(x ,..., x ) ∈ K ,and A ∈ K : 1 m j j=1 j j t t [l] g:(x ,..., x ) → f (A · x , ··· ,A · x )isin C , 1 m 1 m 1 m where the juxtaposition a f denotes the Hadamard product of the two vectors (i.e. the component-wise product (a ,...,a ) · (b ,...,b )=(a b ,..., a b )). 1 n 1 n 1 1 n n Clonoids naturally appear in the study of promise constraint satisfac- tion problems (PCSPs). These problems are investigated, e.g., in [4], and in [5] clonoid theory has been used to provide an algebraic approach to PCSPs. In [14] A. Sparks investigate the number of clonoids for a finite set A and finite algebra B closed under the operations of B.In[8]S.Kreineckerchar- acterized linearly closed clonoids on Z , where p is a prime. Furthermore, a description of the set of all (F, K)-linearly closed clonoids is a useful tool to investigate (polynomial) clones on Z , where n is a product of distinct primes or to represent polynomial functions of semidirect products of groups. In [6] there is a complete description of the structure of all (F, K)-linearly closed clonoids in case F and K are fields and the results we will present are a generalization of this description. The main result of this paper (Theorem 1.2) states that every (F, K)- linearly closed clonoid is generated by its subset of unary functions. m s Theorem 1.2. Let K = F , F = F be products of fields such that q p i i i=1 i=1 |K| and |F| are coprime. Then every (F, K)-linearly closed clonoid is generated by a set of unary functions and thus there are finitely many distinct (F, K)- linearly closed clonoids. The proof of this result is given in Section 3. From this follows that under the assumptions of Theorem 1.2 we can bound the cardinality of the lattice of all (F, K)-linearly closed clonoids. Closed sets of functions Page 3 of 12 61 Furthermore, in Section 4 we find a description of the lattice of all (F, K)- linearly closed clonoids as the direct product of the lattices of all F [K ]- K × submodules of F , where K is the multiplicative monoid of K = F . p i=1 i Moreover, we provide a concrete bound for the cardinality of the lattice of all (F, K)-linearly closed clonoids. s m Theorem 1.3. Let F = F and K = F be products of finite fields p q i j i=1 j=1 such that |K| and |F| are coprime. Then the cardinality of the lattice of all (F, K)-linearly closed clonoids L(F, K) is bounded by: |L(F, K)|≤ , i=1 1≤r≤n where n = q and j=1 n−h+i n q − 1 h q − 1 i=1 with q ∈ N\{1}. 2. Preliminaries and notation We use boldface letters for vectors, e.g., u =(u ,...,u ) for some n ∈ N. 1 n Moreover, we will use v , u  for the scalar product of the vectors v and u . Let f be an n-ary function from an additive group G to a group G .Wesay 1 2 that f is 0-preserving if f (0 ,..., 0 )=0 . A non-trivial (F, K)-linearly G G G 1 1 2 closed clonoids is the set of all 0-preserving finitary functions from K to F. The (F, K)-linearly closed clonoids form a lattice with the intersection as meet and the (F, K)-linearly closed clonoid generated by the union as join. The top element of the lattice is the (F, K)-linearly closed clonoid of all functions and the bottom element consists of only the constant zero functions. We write Clg(S) for the (F, K)-linearly closed clonoid generated by a set of functions S. In order to prove Theorem 1.2 we introduce the definition of 0-absorbing function. This concept is slightly different from the one in [1] since we consider the source set to be split into a product of sets. Nevertheless, some of the techniques in [1] can be used also with our definition of 0-absorbing function. Let A ,...,A be sets, let 0 ∈ A , and let J ⊆ [m]. For all a = 1 m A i m m (J) (J) (a ,...,a ) ∈ A we define a ∈ A by a = a for i ∈ J and 1 m i i i i=1 i=1 i (J) (a ) =0 for i ∈ [m]\J . i A Let A ,...,A be sets, let 0 ∈ A ,let G = G, +, −, 0  be an abelian 1 m A i G group, let f : A → G, and let I ⊆ [m]. By Dep(f ) we denote {i ∈ i=1 [m] | f depends on its ith set argument}. We say that f is 0 -absorbing in its jth argument if for all a =(a ,...,a ) ∈ A with a =0 we have 1 m i j A i=1 f (a)=0 . We say that f is 0-absorbing in I if Dep(f ) ⊆ I and for every i ∈ I f is 0 -absorbing in its ith argument. Using the same proof of [1, Lemma 3] we can find an interesting property of 0-absorbing functions. 61 Page 4 of 12 S. Fioravanti Algebra Univers. Lemma 2.1. Let A ,...,A be sets, let 0 be an element of A for all i ∈ [m]. 1 m A i Let B = B, +, −, 0  be an abelian group, and let f : A → B.Then G i i=1 there is exactly one sequence {f } of functions from A to B such I i I⊆[m] i=1 that for each I ⊆ [m], f is 0-absorbing in I and f = f . Furthermore, I I I⊆[m] i=1 each function f lies in the subgroup F of B that is generated by the (J) functions x → f (x ),where J ⊆ [m]. Proof. The proof is essentially the same of [1, Lemma 3] substituting A with A . We define f by recursion on |I|. We define f (a):= f (0 ,..., 0 ) i I ∅ A A 1 m i=1 and for all I = ∅ and a ∈ A and f by: i I i=1 (I) f (a):= f (a ) − f (a ), (2.1) I J J ⊂I for all a ∈ A . i=1 Furthermore, as in [3], we can see that the component f satisfies f (a ) I I |I|+|J | (J) = (−1) f (a ). From now on we will not specify the element that J ⊆I the functions absorb since it will always be the 0 of a finite field. 3. Unary generators of (F, K)-linearly closed clonoid In this section our aim is to find an analogon of [6, Theorem 4.2] for a generic (F, K)-linearly closed clonoid C, which will allow us to generate C with a set of unary functions. In general we will see that it is the unary part of an (F, K)- linearly closed clonoid that determines the clonoid. To this end we shall show the following lemmata. We denote by e =(1, 0,... , 0) the first member of n k the canonical basis of F as a vector-space over F .Let f : F → F . q p q i q i i=1 i s s Let s ≤ m and let K = F . Then we denote by f | : F → F the q K p i q i=1 i=1 i function such that f | (x ,..., x )= f (x ,..., x , 0,..., 0). K 1 s 1 s Lemma 3.1. Let f, g : F → F be functions, and let b ,..., b be such p 1 m i=1 i that b ∈ F \{(0,... , 0)} for all i ∈ [m]. Assume that f (λ b ,...,λ b )= i 1 1 m m n n F F q q 1 m g(λ e ,...,λ e ), for all λ ∈ F ,...,λ ∈ F ,and f (x)= g(y)= 1 m 1 q m q 1 1 1 m m m 0 for all x ∈ F \{(λ b ,... ,λ b ) | (λ ,...,λ ) ∈ F } and 1 1 m m 1 m q q i i=1 i i=1 n n F F m q q m n 1 m y ∈ F \{(λ e ,...,λ e ) | (λ ,..., λ ) ∈ F }.Then f ∈ 1 m 1 m q q 1 1 i i=1 i i=1 Clg({g}). Proof. For j ≤ m let A be any invertible n × n-matrix over K such that j j A b = e . Then is straightforward to check that f (x ,..., x )= g(Ax , j j 1 m 1 ...,Ax ). Lemma 3.2. Let q ,...,q and p be powers of primes and let K = F . 1 m q i=1 i Let h ≤ m and let K = F .Let C be an (F , K)-linearly closed clonoid 1 q p i=1 i and let C | := {g |∃g ∈ C : g | = g}. K K 1 1 s [1] [s] Let Dep(f)=[h] and let f : F → F .Then f ∈ Clg(C ) if and only i=1 i [1] [s] if f | ∈ Clg(C | ) . 1 Closed sets of functions Page 5 of 12 61 [1] [1] Proof. It is clear that if f ∈ Clg(C ) then f | ∈ Clg(C | ), simply restrict- 1 K ing to K all the unary generators of f . Conversely, let S be a set of unary [1] generators of f | .Let S ⊆ C be defined by S := g |∃g ∈ S : g(x ,...,x , 0,..., 0) = g (x ,...,x ), 1 h 1 h for all (x ,...,x ) ∈ F . 1 h q i=1 From Dep(f)=[h] follows that S is a set of unary generators of f . Lemma 3.3. Let q ,...,q and p be powers of primes with q and p 1 m i i=1 coprime. Let K = F .Let C be an (F , K)-linearly closed clonoid, let q p i=1 i [1] k g ∈ C be 0-absorbing in [m],and let t : F → F be defined by: k q p i=1 i 1 m t (λ e ,...,λ e )= g(λ ,...,λ ) for all (λ ,...,λ ) ∈ F k 1 m 1 m 1 m q 1 1 i=1 t (x)=0 m m k k F F q q k 1 m for all x ∈ F \ (λ e ,...,λ e ) | (λ ,...,λ ) ∈ F . 1 m 1 m q q 1 1 i i=1 i=1 Then t is 0-absorbing in [m],with A = F and 0 =(0 ,..., 0 ).Fur- k i A F F q i q q i i [1] thermore, t ∈ Clg(C ) for all k ∈ N. Proof. Since g is 0-absorbing in [m] then also t is 0-absorbing in [m]. Moreover [1] ,we prove that t ∈ Clg(C ) by induction on k. [1] Case k =1: if k = 1, then t = g is a unary function of C . [1] Case k> 1: we assume that t ∈ Clg(C ). For all 1 ≤ i ≤ m we define the k−1 [k] [k] k k−1 two sets of mappings T and R from F to F by: i i q q i i [k] T := {u :(x ,...,x ) → (x − ax ,x ...,x ) | a ∈ F } a 1 k 1 2 3 k q i i [k] R := {w :(x ,...,x ) → (ax ,x ...,x ) | a ∈ F \{0}}. a 1 k 2 3 k q i i [k] [k] [k] m [k] [k] Let P := T ∪R . Furthermore, we define the function c : P → N i i i i=1 i by: m [k] 0if h ∈ T , [k] i=1 i c (h)= m [k] 1if h ∈ R . i=1 i Let us define the function r : F → F by: k p i=1 i r (x ,..., x ) k 1 m m [k] c (h ) i=1 = (−1) t (h (x ),,...,h (x )), (3.1) k−1 1 1 m m [k] [k] h ∈P ,...,h ∈P 1 m m for all x ∈ F . i 61 Page 6 of 12 S. Fioravanti Algebra Univers. Claim: r (x ,..., x )= q · t (x ,..., x ) for all (x ,..., x ) ∈ k 1 m i k 1 m 1 m i=1 i=1 q Subcase ∃i ∈ [m], 3 ≤ j ≤ k with (x ) =0: i j By definition of t ,wecan seethatin(3.1) every summand vanishes k−1 if there exist i ∈ [m] and 3 ≤ j ≤ k with (x ) =0. Thus r (x ,..., x )= i j k 1 m q · t (x ,..., x ) = 0 in this case. i k 1 m i=1 Subcase ∃l ∈ [m] with (x ) =0 and (x ) = 0 for all i ∈ [m], 3 ≤ j ≤ k: l 2 i j We prove that r (x ,..., x ) = 0. We can see that for all (x ,x ) ∈ F × k 1 m 1 2 q F \{0} and for all b ∈ F \{0}, there exists a ∈ F such that bx = x − ax , q q q 2 1 2 l l l −1 and clearly a = x x − b. Conversely, for all (x ,x ) ∈ F × F \{0} and for 1 1 2 q q 2 l l −1 all a ∈ F \{x x } there exists b ∈ F \{0} such bx = x − ax , and clearly q 1 q 2 1 2 l 2 l −1 b = x x − a. [k] With this observation we can see that for all h ∈ P with i ∈ [m]\{l} and for all (x ,..., x ) ∈ F with (x ) = x and (x ) = x we have 1 m l 1 1 l 2 2 i=1 i −1 that if a = x x then: t (h (x ),...,h (x ),u (x ),h (x ),...,h (x )) k−1 1 1 l−1 l−1 a l l+1 l+1 m m −1 = t (h (x ),...,h (x ),w (x ),h (x ),...,h (x )) k−1 1 1 l−1 l−1 l l+1 l+1 m m x x −a [k] [k] where u ∈ T and w −1 ∈ R . Thus they produce summands with l x x −a l −1 different signs in (3.1). Moreover, if a = x x , then t (h (x ),...,h (x ),u (x ),h (x ),...,h (x )) k−1 1 1 l−1 l−1 a l l+1 l+1 m m = t (h (x ),...,h (x ), 0 k−1,h (x ),...,h (x )) = 0, k−1 1 1 l−1 l−1 l+1 l+1 m m since t is 0-absorbing in [m]. This implies that all the summands of r are k−1 k cancelling if (x ) =0. Thus r (x ,..., x )= q · t (x ,..., x )=0 in l 2 k 1 m i k 1 m i=1 this case. 1 m Subcase (x ,..., x )=(λ e ,...,λ e ) for some (λ ,...,λ ) ∈ 1 m 1 m 1 m 1 1 F : i=1 i We can observe that: t (h (x ),...,h (x ),h (λ e ),h (x ),...,h (x )) = 0 k−1 1 1 l−1 l−1 l l l+1 l+1 m m = t (h (x ),...,h (x ), 0 k−1,h (x ),...,h (x )) = 0, k−1 1 1 l−1 l−1 l+1 l+1 m m l Closed sets of functions Page 7 of 12 61 [k] for all h ∈ P with i ∈ [m]\{l}, for all l ≤ m, λ ∈ F , x ∈ F ,and i l q i i l q [k] h ∈ R , since t is 0-absorbing in [n]. Thus we can observe that: l k−1 1 m r (λ e ,...,λ e ) k 1 m 1 1 k k m [k] F F c (h ) q q i 1 m i=1 = (−1) t (h (λ e ),...,h (λ e )) k−1 1 1 m m 1 1 [k] h ∈P k k m [k] F F c (h ) q q i 1 m i=1 = (−1) t (h (λ e ),...,h (λ e )) k−1 1 1 m m 1 1 [k] h ∈T k k F F q q 1 m = t (h (λ e ),...,h (λ e )) k−1 1 1 m m 1 1 [k] h ∈T k−1 k−1 1 m = t (λ e ,...,λ e ) k−1 1 m 1 1 [k] h ∈T k−1 k−1 1 m = q · t (λ e ,...,λ e ) i k−1 1 m 1 1 i=1 k k F F q q 1 m = q · t (λ e ,...,λ e ). i k 1 m 1 1 i=1 Thus r = q · t . k i k i=1 Because of (3.1) and the inductive hypothesis, we have r ∈ Clg({t }) k k−1 m m [1] [1] ⊆ Clg(C ). Thus q t ∈ Clg(C ). Since q = 0 modulo p we have i k i i=1 i=1 [1] that t ∈ Clg(C ) and this concludes the induction proof. Lemma 3.4. Let q ,...,q and p be powers of primes with q and p co- 1 m i i=1 prime and let K = F .Let C be an (F , K)-linearly closed clonoid, let q p i=1 i [1] I ⊆ [m] and let f ∈ C be 0-absorbing in I.Then f ∈ Clg(C ). Proof. Let K = F and let C := {g |∃g ∈ C : g | = g}. By Lemma 1 q 1 K i∈I i 1 [1] [1] 3.2 f ∈ Clg(C ) if and only if f | ∈ Clg(C | ) and we observe that f | K K 1 K 1 is 0-absorbing in I . Thus without loss of generality we fix I =[m]. The strategy is to interpolate f in all the distinct products of lines of the form {(λ b ,...,λ b ) | (λ ,...,λ ) ∈ F , b ∈ F \{(0,..., 0)}. To this 1 1 m m 1 m q i i=1 i q end let R = {L | 1 ≤ j ≤ (q − 1)/(q − 1) = s} be the set of all s distinct j i i=1 i products of lines of F and let l ∈ F be such that (l ,..., l ) q (i,j) (1,j) (m,j) i=1 i q generates the products of m lines L ,for 1 ≤ j ≤ s,1 ≤ i ≤ m. For all 1 ≤ j ≤ s,let f : F → F be defined by: L p j i=1 q f (λ l ,...,λ l )= f (λ l ,...,λ l ) L 1 (1,j) m (m,j) 1 (1,j) m (m,j) m m for (λ ,...,λ ) ∈ F and f (x ) = 0 for all x ∈ F \{(λ l , 1 m q L 1 (1,j) i=1 i j i=1 q ...,λ l ) | (λ ,...,λ ) ∈ F }. m (m,j) 1 m q i=1 i Claim 1: f = f . j=1 j 61 Page 8 of 12 S. Fioravanti Algebra Univers. Since f is 0-absorbing in [m]wehavethat: f (λ l ,...,λ l )= f (λ l ,...,λ l ) L 1 (1,z) m (m,z) L 1 (1,z) m (m,z) j z j=1 = f (λ l ,...,λ l ) 1 (1,z) m (m,z) for all (λ ,...,λ ) ∈ F and z ∈ [s], since for all j ,j ∈ [s], L 1 m q 1 2 j i 1 i=1 and L intersect only in points of the form (x ,..., x ) ∈ F with j 1 m 2 q i=1 i x =(0,..., 0) for some i ∈ [m]. Let 1 ≤ j ≤ s and let g : F → F be a function such that: q p i=1 i f (λ l ,...,λ l )= g(λ ,...,λ )= f (λ l ,...,λ l ) L 1 (1,j) m (m,j) 1 m 1 (1,j) m (m,j) [1] for all (λ ,...,λ ) ∈ F . Then g ∈ C . 1 m q i=1 [1] Claim 2: f ∈ Clg(C ) for all L ∈ R. L j We can observe that f (λ l ,...,λ l )= g(λ ,...,λ ) for all L 1 (1,j) m (m,j) 1 m (λ ,..., λ ) ∈ F ,and f (x ,..., x ) = 0 for all (x ,..., x ) ∈ 1 m q L 1 m 1 m i=1 i j m m F \{(λ l , ...,λ l ) | (λ ,...,λ ) ∈ F }. Furthermore, 1 (1,j) m (m,j) 1 m q q i i=1 i i=1 [1] g is 0-absorbing in [m]. By Lemmata 3.1 and 3.3, f ∈ Clg(C ), which [1] concludes the proof of f ∈ Clg(C ). We are now ready to prove that an (F, K)-linearly closed clonoid C is generated by its unary part. Theorem 3.5. Let q ,...,q and p be powers of primes with q and p 1 m i i=1 coprime and let K = F . Then every (F , K)-linearly closed clonoid C is q p i=1 i [1] generated by its unary functions. Thus C = Clg(C ). Proof. The inclusion ⊇ is obvious. For the other inclusion let C be an (F , K)- linearly closed clonoid and let f be an n-ary function in C. By Lemma 2.1 with A = F and 0 =(0 ,..., 0 ), f can be split in the sum of n-ary functions i A F F q i q q i i i f such that for each I ⊆ [m], f is 0-absorbing in I. Furthermore, each I I I⊆[m] function f lies in the subgroup F of F that is generated by the functions (I) x → f (x ), where I ⊆ [m] and thus each summand f is in C. By Lemma [1] [1] 3.4 each of these summands is in Clg(C ). and thus f ∈ Clg(C ). The next corollary of Theorem 3.5 and the following theorem tell us that there are only finitely many distinct (F, K)-linearly closed clonoids. Corollary 3.6. Let q ,...,q and p be powers of primes with q and p 1 m i i=1 coprime and let K = F .Let C and D be two (F , K)-linearly closed q p i=1 [1] [1] clonoids. Then C = D if and only if C = D . Let us denote by L(F, K) the lattice of all (F, K)-linearly closed clonoids. We define the functions ρ : L(F, K) →L(F , K) such that for all 1 ≤ i ≤ s i p and for all C ∈L(F, K): ρ (C):= {f | there exists g ∈ C : f = π ◦ g}, (3.2) where with π we denote the projection over the i-th component of the product of fields F. Closed sets of functions Page 9 of 12 61 s m Theorem 3.7. Let F = F and K = F be products of finite fields. p q i i i=1 i=1 Then the lattice of all (F, K)-linearly closed clonoids is isomorphic to the direct product of the lattices of all (F , K)-linearly closed clonoids with 1 ≤ i ≤ s. Proof. Let us define the function ρ : L(F, K) → L(F , K) such that i=1 i ρ(C):=(ρ (C),...,ρ (C)). Clearly ρ is well-defined. Conversely, let ψ : L 1 s i=1 (F , K) →L(F, K) be defined by: [k] [k] ψ(C ,...,C )= {f : x → (f (x ),...,f (x )) | f ∈ C ,...,f ∈ C }. 1 s 1 s 1 s 1 s k∈N From this definition it is clear that ψ is well defined. Furthermore, ρψ(C ,...,C )=(C ,...,C ) 1 s 1 s and C ⊆ ψρ(C) for all (C ,...,C ) ∈ L(F , K)and C ∈L(F, K). 1 s p i=1 To prove that C ⊇ ψρ(C)let f ∈ ψρ(C). Then there exists (f ,...,f ) ∈ 1 s ρ(C) such that f = π ◦ f for all i ∈ [s]. By definition of ρ, there exist g ,...,g ∈ C such that f = π ◦ g for all i ∈ [s]. Let a ∈ F be such 1 s i i i that a (j)=0 for j = i and a (i) = 1. It is easy to check that the function i i f = a g = f and thus f ∈ C. i i i=1 Hence ρ is a lattice isomorphism. s m Proof of Theorem 1.2. Let F = F and K = F be products of p q i i i=1 i=1 finite with |K| and |F| coprime. Let C ∈L(F, K). By Theorem 3.7 C is uniquely determined by its projections C = ρ (C),...,C = ρ (C) where ρ is defined 1 1 s s i in (3.2). By Theorem 3.5 we have that for all i ∈ [s]every (F , K)-linearly [1] closed clonoid C is uniquely determined by its unary part C .Thus C is [1] uniquely determined by its unary part C . 4. The lattice of all (F, K)-linearly closed clonoids In this section we characterize the structure of the lattice L(F, K) of all (F, K)- linearly closed clonoids through a description of their unary parts. Let F = s m F and K = F be products of finite fields such that |K| and |F| p q i=1 i j=1 j are coprime numbers. We will see that L(F, K) is isomorphic to the product of the lattices × K × of all F [K ]-submodules of F , where K is the multiplicative monoid of i p K = F . In order to characterize the lattice of all (F, K)-linearly closed i=1 i clonoids we need the definition of monoid ring. Definition 4.1. Let M, · be a commutative monoid and let R, +,  be a commutative ring with identity. Let S := {f ∈ R | f (a) = 0 for only finitely many a ∈ M }. We define the monoid ring of M over R as the ring (S, +, ·), where + is the point-wise addition of functions and the multiplication is defined as f · g : R → M which maps each m ∈ M into: f (m )g(m ). 1 2 m ,m ∈M,m m =m 1 2 1 2 61 Page 10 of 12 S. Fioravanti Algebra Univers. We denote by R[M ] the monoid ring of M over R. Following the notation in [3] for all a ∈ A we define τ to be the element of R with τ (a)=1and a a τ (M \{a})= {0}. We observe that for all f ∈ R[M ] there is an r ∈ R such that f = r τ and that we can multiply such expressions with the rule a a a∈M τ · τ = τ . a b ab Definition 4.2. Let M be a commutative monoid and let R be a commutative ring. We denote by R the R[M ]-module with the action ∗ defined by: (τ ∗ f )(x)= f (ax), for all a ∈ M and f ∈ R . Let K be the multiplicative monoid of K = F . We can observe i=1 i × K K that V is an F [K ]-submodule of F if and only if it is a subspace of F p p satisfying (x ,...,x ) → f (a x ,...,a x ) ∈ V, (4.1) 1 m 1 1 m m for all f ∈ V and (a ,...,a ) ∈ F . Clearly the following lemma holds. 1 m q i=1 i Lemma 4.3. Let p, q ,...q be powers of primes and let K = F .Let 1 m q i=1 V ⊆ F .Then V is the unary part of an (F , K)-linearly closed clonoid if and × K only if is an F [K ]-submodule of F . Together with Theorem 3.7 this immediately yields the following. m s Corollary 4.4. Let K = F and F = F be products of finite fields q p i i i=1 i=1 [1] such that |K| and |F| are coprime. Then the function π that sends an (F, K)- linearly closed clonoid to its unary part is an isomorphism between the lattice of all (F, K)-linearly closed clonoids and the direct product of the lattices of all × K F [K ]-submodules of F . i p With the same strategy of [6, Lemma 5.6] we obtain the following Lemma. m s Lemma 4.5. Let K = F and F = F be products of finite fields q p i=1 i i=1 i such that |K| and |F| are coprime. Then every (F, K)-linearly closed clonoid is finitely related. The next step is to characterize the lattice of all F [K ]-submodules of K × K F . To this end we observe that V is an F [K ]-submodule of F if and only p p if is a subspace of F satisfying (4.1). We can provide a bound for the lattice of all (F, K)-linearly closed clonoids given by the number of subspaces of F . Remark 4.6. It is a well-known fact in linear algebra that the number of k- dimensional subspaces of an n-dimensional vector space V over a finite field F is the Gaussian binomial coefficient: n−k+i n q − 1 = . (4.2) k q − 1 i=1 Closed sets of functions Page 11 of 12 61 From this remark we directly obtain the bound of Theorem 1.3. In order to determine the exact cardinality of the lattice of all (F, K)-linearly closed clonoids we have to deal with the problem to find the F [K ]-submodules of F . We will not study this problem here because we think that this is an interesting problem that deserves an own research. Acknowledgements The author thanks Erhard Aichinger, who inspired this paper, Erkko Lehtonen who reviewed my Ph.D. thesis, and Sebastian Kreinecker for many hours of fruitful discussions. The author thanks the referees for their useful suggestions. Funding Open access funding provided by Johannes Kepler University Linz. Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and re- production in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regu- lation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons. org/licenses/by/4.0/. Publisher’s Note Springer Nature remains neutral with regard to jurisdic- tional claims in published maps and institutional affiliations. References [1] Aichinger, E.: Solving systems of equations in supernilpotent algebras. In: 44th International Symposium on Mathematical Foundations of Computer Science, LIPIcs. Leibniz Int. Proc. Inform, 138, Art. No. 72, 9, Schloss Dagstuhl. Leibniz- Zent. Inform., Wadern (2019) [2] Aichinger, E., Mayr, P.: Finitely generated equational classes. J. Pure Appl. Algebra 220(8), 2816–2827 (2016) [3] Aichinger, E., Moosbauer, J.: Chevalley-Warning type results on abelian groups. J. Algebra 569, 30–66 (2021) [4] Brakensiek, J., Guruswami, V.: Promise constraint satisfaction structure the- ory and a symmetric Boolean dichotomy. In: Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms ( SODA’18), pp. 1782– 1801. SIAM, Philadelphia (2018) 61 Page 12 of 12 S. Fioravanti Algebra Univers. [5] Bul´ ın, J., Krokhin, A., Oprˇ sal, J.: Algebraic approach to promise constraint sat- isfaction. In: Proceedings of the Annual ACM Symposium on Theory of Com- puting (STOC ’19), pp. 602–613. ACM, New York (2019) [6] Fioravanti, S.: Closed sets of finitary functions between finite fields of coprime order. Algebra Universalis 81, 52 (2020) [7] Harnau, W.: Ein verallgemeinerter Relationenbegriff fur ¨ die Algebra der mehrw- ertigen Logik. I. Grundlagen Rostock. Math. Kolloq. 28, 5–17 (1985). (German) [8] Kreinecker, S.: Closed function sets on groups of prime order. J. Multiple-Valued Logic Soft Comput. 33(1–2), 51–74 (2019) [9] Krokhin, A., Bulatov, A.A., Jeavons, P.: The complexity of constraint satisfac- tion: an algebraic approach. In: Structural Theory of Automata, Semigroups, and Universal Algebra. NATO Sci. Ser. II Math. Phys. Chem., vol. 207, pp. 181–213. Springer, Dordrecht (2005) [10] Lehtonen, E.: Closed classes of functions, generalized constraints, and clusters. Algebra Universalis 63(2–3), 203–234 (2010) [11] P¨ oschel, R., Kaluˇ znin, L.A.: Funktionen- und Relationenalgebren. Mathematis- che Monographien [Mathematical Monographs], vol. 15. VEB Deutscher Verlag der Wissenschaften, Berlin (1979) [12] Post, E.L.: The two-valued iterative systems of mathematical logic. Ann. of Math. Stud., vol. 5. Princeton University Press, Princeton (1941) [13] Rosenberg, I.: Maximal clones on algebras A and A . Rend. Circ. Mat. Palermo 2(18), 329–333 (1969) [14] Sparks, A.: On the number of clonoids. Algebra Universalis 80(4), 53 (2019) [15] Szendrei, A.: Clones in universal algebra. In: S´ eminaire de Math´ ematiques Sup´ erieures. Seminar on Higher Mathematics, vol. 99. Presses de l’Universit´ e de Montr´ eal, Montreal (1986) Stefano Fioravanti Institut fur ¨ Algebra Johannes Kepler Universit¨ at Linz 4040 Linz Austria e-mail: stefano.fioravanti66@gmail.com Received: 4 September 2020. Accepted: 11 August 2021. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png algebra universalis Springer Journals

Closed sets of finitary functions between products of finite fields of coprime order

Fioravanti, Stefano
algebra universalis , Volume 82 (4) – Nov 1, 2021
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Algebra Univers. (2021) 82:61 c 2021 The Author(s) Algebra Universalis https://doi.org/10.1007/s00012-021-00748-z Closed sets of finitary functions between products of finite fields of coprime order Stefano Fioravanti Abstract. We investigate the finitary functions from a finite product of m n finite fields F = K to a finite product of finite fields F = F, q p j=1 j i=1 i where |K| and |F| are coprime. An (F, K)-linearly closed clonoid is a subset of these functions which is closed under composition from the right and from the left with linear mappings. We give a characterization of these × K × subsets of functions through the F [K ]-submodules of F ,where K is p p the multiplicative monoid of K = F . Furthermore we prove that i=1 i each of these subsets of functions is generated by a set of unary functions and we provide an upper bound for the number of distinct (F, K)-linearly closed clonoids. Mathematics Subject Classification. 08A40. Keywords. Clonoids, Clones. 1. Introduction Since P. Hall’s abstract definition of a clone the problem to describe sets of finitary functions from a set A to a set B which satisfy some closure properties has been a fruitful branch of research. E. Post’s characterization of all clones on a two-element set [12] can be considered as a foundational result in this field, which was developed further, e.g., in [13, 11, 15, 10]. Starting from [9], 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 a finite product m n of finite fields F = K to a finite product of finite fields F = F, q p j=1 j i=1 i where |K| and |F| are coprime. The sets of functions we are interested in are closed under composition from the left and from the right with linear mappings. Thus we consider sets of functions with different domains and codomains; such Presented by R. P¨ oschel. The research was supported by the Austrian Science Fund (FWF): P29931. 0123456789().: V,-vol 61 Page 2 of 12 S. Fioravanti Algebra Univers. sets are called clonoids and are investigated, e.g., in [2]. Let B be an algebra, and let A be a non-empty set. For a subset C of B and k ∈ N,welet n∈N [k] A C := C ∩ B . According to Definition 4.1of[2] we call C a clonoid with source set A and target algebra B if [k] A (1) for all k ∈ N: C is a subuniverse of B ,and k [k] (2) for all k, n ∈ N, for all (i ,...,i ) ∈{1,...,n} , and for all c ∈ C ,the 1 k n  [n] function c : A → B with c (a ,...,a ):= c(a ,...,a ) lies in C . 1 n i i 1 k By (1) every clonoid is closed under composition with operations of B on the left. In particular we are dealing with those clonoids whose target algebra is the ring F that are closed under composition with linear mappings i=1 i from the right side. m s Definition 1.1. Let m, s ∈ N and let K = K , F = F be prod- j i j=1 i=1 ucts of fields. An (F, K)-linearly closed clonoid is a non-empty subset C of m k j=1 j F with the following properties: k∈N i=1 i [n] (1) for all n ∈ N, a , b ∈ F ,and f, g ∈ C : i=1 [n] a f + bg ∈ C ; n×l [n] l (2) for all l, n ∈ N, f ∈ C ,(x ,..., x ) ∈ K ,and A ∈ K : 1 m j j=1 j j t t [l] g:(x ,..., x ) → f (A · x , ··· ,A · x )isin C , 1 m 1 m 1 m where the juxtaposition a f denotes the Hadamard product of the two vectors (i.e. the component-wise product (a ,...,a ) · (b ,...,b )=(a b ,..., a b )). 1 n 1 n 1 1 n n Clonoids naturally appear in the study of promise constraint satisfac- tion problems (PCSPs). These problems are investigated, e.g., in [4], and in [5] clonoid theory has been used to provide an algebraic approach to PCSPs. In [14] A. Sparks investigate the number of clonoids for a finite set A and finite algebra B closed under the operations of B.In[8]S.Kreineckerchar- acterized linearly closed clonoids on Z , where p is a prime. Furthermore, a description of the set of all (F, K)-linearly closed clonoids is a useful tool to investigate (polynomial) clones on Z , where n is a product of distinct primes or to represent polynomial functions of semidirect products of groups. In [6] there is a complete description of the structure of all (F, K)-linearly closed clonoids in case F and K are fields and the results we will present are a generalization of this description. The main result of this paper (Theorem 1.2) states that every (F, K)- linearly closed clonoid is generated by its subset of unary functions. m s Theorem 1.2. Let K = F , F = F be products of fields such that q p i i i=1 i=1 |K| and |F| are coprime. Then every (F, K)-linearly closed clonoid is generated by a set of unary functions and thus there are finitely many distinct (F, K)- linearly closed clonoids. The proof of this result is given in Section 3. From this follows that under the assumptions of Theorem 1.2 we can bound the cardinality of the lattice of all (F, K)-linearly closed clonoids. Closed sets of functions Page 3 of 12 61 Furthermore, in Section 4 we find a description of the lattice of all (F, K)- linearly closed clonoids as the direct product of the lattices of all F [K ]- K × submodules of F , where K is the multiplicative monoid of K = F . p i=1 i Moreover, we provide a concrete bound for the cardinality of the lattice of all (F, K)-linearly closed clonoids. s m Theorem 1.3. Let F = F and K = F be products of finite fields p q i j i=1 j=1 such that |K| and |F| are coprime. Then the cardinality of the lattice of all (F, K)-linearly closed clonoids L(F, K) is bounded by: |L(F, K)|≤ , i=1 1≤r≤n where n = q and j=1 n−h+i n q − 1 h q − 1 i=1 with q ∈ N\{1}. 2. Preliminaries and notation We use boldface letters for vectors, e.g., u =(u ,...,u ) for some n ∈ N. 1 n Moreover, we will use v , u  for the scalar product of the vectors v and u . Let f be an n-ary function from an additive group G to a group G .Wesay 1 2 that f is 0-preserving if f (0 ,..., 0 )=0 . A non-trivial (F, K)-linearly G G G 1 1 2 closed clonoids is the set of all 0-preserving finitary functions from K to F. The (F, K)-linearly closed clonoids form a lattice with the intersection as meet and the (F, K)-linearly closed clonoid generated by the union as join. The top element of the lattice is the (F, K)-linearly closed clonoid of all functions and the bottom element consists of only the constant zero functions. We write Clg(S) for the (F, K)-linearly closed clonoid generated by a set of functions S. In order to prove Theorem 1.2 we introduce the definition of 0-absorbing function. This concept is slightly different from the one in [1] since we consider the source set to be split into a product of sets. Nevertheless, some of the techniques in [1] can be used also with our definition of 0-absorbing function. Let A ,...,A be sets, let 0 ∈ A , and let J ⊆ [m]. For all a = 1 m A i m m (J) (J) (a ,...,a ) ∈ A we define a ∈ A by a = a for i ∈ J and 1 m i i i i=1 i=1 i (J) (a ) =0 for i ∈ [m]\J . i A Let A ,...,A be sets, let 0 ∈ A ,let G = G, +, −, 0  be an abelian 1 m A i G group, let f : A → G, and let I ⊆ [m]. By Dep(f ) we denote {i ∈ i=1 [m] | f depends on its ith set argument}. We say that f is 0 -absorbing in its jth argument if for all a =(a ,...,a ) ∈ A with a =0 we have 1 m i j A i=1 f (a)=0 . We say that f is 0-absorbing in I if Dep(f ) ⊆ I and for every i ∈ I f is 0 -absorbing in its ith argument. Using the same proof of [1, Lemma 3] we can find an interesting property of 0-absorbing functions. 61 Page 4 of 12 S. Fioravanti Algebra Univers. Lemma 2.1. Let A ,...,A be sets, let 0 be an element of A for all i ∈ [m]. 1 m A i Let B = B, +, −, 0  be an abelian group, and let f : A → B.Then G i i=1 there is exactly one sequence {f } of functions from A to B such I i I⊆[m] i=1 that for each I ⊆ [m], f is 0-absorbing in I and f = f . Furthermore, I I I⊆[m] i=1 each function f lies in the subgroup F of B that is generated by the (J) functions x → f (x ),where J ⊆ [m]. Proof. The proof is essentially the same of [1, Lemma 3] substituting A with A . We define f by recursion on |I|. We define f (a):= f (0 ,..., 0 ) i I ∅ A A 1 m i=1 and for all I = ∅ and a ∈ A and f by: i I i=1 (I) f (a):= f (a ) − f (a ), (2.1) I J J ⊂I for all a ∈ A . i=1 Furthermore, as in [3], we can see that the component f satisfies f (a ) I I |I|+|J | (J) = (−1) f (a ). From now on we will not specify the element that J ⊆I the functions absorb since it will always be the 0 of a finite field. 3. Unary generators of (F, K)-linearly closed clonoid In this section our aim is to find an analogon of [6, Theorem 4.2] for a generic (F, K)-linearly closed clonoid C, which will allow us to generate C with a set of unary functions. In general we will see that it is the unary part of an (F, K)- linearly closed clonoid that determines the clonoid. To this end we shall show the following lemmata. We denote by e =(1, 0,... , 0) the first member of n k the canonical basis of F as a vector-space over F .Let f : F → F . q p q i q i i=1 i s s Let s ≤ m and let K = F . Then we denote by f | : F → F the q K p i q i=1 i=1 i function such that f | (x ,..., x )= f (x ,..., x , 0,..., 0). K 1 s 1 s Lemma 3.1. Let f, g : F → F be functions, and let b ,..., b be such p 1 m i=1 i that b ∈ F \{(0,... , 0)} for all i ∈ [m]. Assume that f (λ b ,...,λ b )= i 1 1 m m n n F F q q 1 m g(λ e ,...,λ e ), for all λ ∈ F ,...,λ ∈ F ,and f (x)= g(y)= 1 m 1 q m q 1 1 1 m m m 0 for all x ∈ F \{(λ b ,... ,λ b ) | (λ ,...,λ ) ∈ F } and 1 1 m m 1 m q q i i=1 i i=1 n n F F m q q m n 1 m y ∈ F \{(λ e ,...,λ e ) | (λ ,..., λ ) ∈ F }.Then f ∈ 1 m 1 m q q 1 1 i i=1 i i=1 Clg({g}). Proof. For j ≤ m let A be any invertible n × n-matrix over K such that j j A b = e . Then is straightforward to check that f (x ,..., x )= g(Ax , j j 1 m 1 ...,Ax ). Lemma 3.2. Let q ,...,q and p be powers of primes and let K = F . 1 m q i=1 i Let h ≤ m and let K = F .Let C be an (F , K)-linearly closed clonoid 1 q p i=1 i and let C | := {g |∃g ∈ C : g | = g}. K K 1 1 s [1] [s] Let Dep(f)=[h] and let f : F → F .Then f ∈ Clg(C ) if and only i=1 i [1] [s] if f | ∈ Clg(C | ) . 1 Closed sets of functions Page 5 of 12 61 [1] [1] Proof. It is clear that if f ∈ Clg(C ) then f | ∈ Clg(C | ), simply restrict- 1 K ing to K all the unary generators of f . Conversely, let S be a set of unary [1] generators of f | .Let S ⊆ C be defined by S := g |∃g ∈ S : g(x ,...,x , 0,..., 0) = g (x ,...,x ), 1 h 1 h for all (x ,...,x ) ∈ F . 1 h q i=1 From Dep(f)=[h] follows that S is a set of unary generators of f . Lemma 3.3. Let q ,...,q and p be powers of primes with q and p 1 m i i=1 coprime. Let K = F .Let C be an (F , K)-linearly closed clonoid, let q p i=1 i [1] k g ∈ C be 0-absorbing in [m],and let t : F → F be defined by: k q p i=1 i 1 m t (λ e ,...,λ e )= g(λ ,...,λ ) for all (λ ,...,λ ) ∈ F k 1 m 1 m 1 m q 1 1 i=1 t (x)=0 m m k k F F q q k 1 m for all x ∈ F \ (λ e ,...,λ e ) | (λ ,...,λ ) ∈ F . 1 m 1 m q q 1 1 i i=1 i=1 Then t is 0-absorbing in [m],with A = F and 0 =(0 ,..., 0 ).Fur- k i A F F q i q q i i [1] thermore, t ∈ Clg(C ) for all k ∈ N. Proof. Since g is 0-absorbing in [m] then also t is 0-absorbing in [m]. Moreover [1] ,we prove that t ∈ Clg(C ) by induction on k. [1] Case k =1: if k = 1, then t = g is a unary function of C . [1] Case k> 1: we assume that t ∈ Clg(C ). For all 1 ≤ i ≤ m we define the k−1 [k] [k] k k−1 two sets of mappings T and R from F to F by: i i q q i i [k] T := {u :(x ,...,x ) → (x − ax ,x ...,x ) | a ∈ F } a 1 k 1 2 3 k q i i [k] R := {w :(x ,...,x ) → (ax ,x ...,x ) | a ∈ F \{0}}. a 1 k 2 3 k q i i [k] [k] [k] m [k] [k] Let P := T ∪R . Furthermore, we define the function c : P → N i i i i=1 i by: m [k] 0if h ∈ T , [k] i=1 i c (h)= m [k] 1if h ∈ R . i=1 i Let us define the function r : F → F by: k p i=1 i r (x ,..., x ) k 1 m m [k] c (h ) i=1 = (−1) t (h (x ),,...,h (x )), (3.1) k−1 1 1 m m [k] [k] h ∈P ,...,h ∈P 1 m m for all x ∈ F . i 61 Page 6 of 12 S. Fioravanti Algebra Univers. Claim: r (x ,..., x )= q · t (x ,..., x ) for all (x ,..., x ) ∈ k 1 m i k 1 m 1 m i=1 i=1 q Subcase ∃i ∈ [m], 3 ≤ j ≤ k with (x ) =0: i j By definition of t ,wecan seethatin(3.1) every summand vanishes k−1 if there exist i ∈ [m] and 3 ≤ j ≤ k with (x ) =0. Thus r (x ,..., x )= i j k 1 m q · t (x ,..., x ) = 0 in this case. i k 1 m i=1 Subcase ∃l ∈ [m] with (x ) =0 and (x ) = 0 for all i ∈ [m], 3 ≤ j ≤ k: l 2 i j We prove that r (x ,..., x ) = 0. We can see that for all (x ,x ) ∈ F × k 1 m 1 2 q F \{0} and for all b ∈ F \{0}, there exists a ∈ F such that bx = x − ax , q q q 2 1 2 l l l −1 and clearly a = x x − b. Conversely, for all (x ,x ) ∈ F × F \{0} and for 1 1 2 q q 2 l l −1 all a ∈ F \{x x } there exists b ∈ F \{0} such bx = x − ax , and clearly q 1 q 2 1 2 l 2 l −1 b = x x − a. [k] With this observation we can see that for all h ∈ P with i ∈ [m]\{l} and for all (x ,..., x ) ∈ F with (x ) = x and (x ) = x we have 1 m l 1 1 l 2 2 i=1 i −1 that if a = x x then: t (h (x ),...,h (x ),u (x ),h (x ),...,h (x )) k−1 1 1 l−1 l−1 a l l+1 l+1 m m −1 = t (h (x ),...,h (x ),w (x ),h (x ),...,h (x )) k−1 1 1 l−1 l−1 l l+1 l+1 m m x x −a [k] [k] where u ∈ T and w −1 ∈ R . Thus they produce summands with l x x −a l −1 different signs in (3.1). Moreover, if a = x x , then t (h (x ),...,h (x ),u (x ),h (x ),...,h (x )) k−1 1 1 l−1 l−1 a l l+1 l+1 m m = t (h (x ),...,h (x ), 0 k−1,h (x ),...,h (x )) = 0, k−1 1 1 l−1 l−1 l+1 l+1 m m since t is 0-absorbing in [m]. This implies that all the summands of r are k−1 k cancelling if (x ) =0. Thus r (x ,..., x )= q · t (x ,..., x )=0 in l 2 k 1 m i k 1 m i=1 this case. 1 m Subcase (x ,..., x )=(λ e ,...,λ e ) for some (λ ,...,λ ) ∈ 1 m 1 m 1 m 1 1 F : i=1 i We can observe that: t (h (x ),...,h (x ),h (λ e ),h (x ),...,h (x )) = 0 k−1 1 1 l−1 l−1 l l l+1 l+1 m m = t (h (x ),...,h (x ), 0 k−1,h (x ),...,h (x )) = 0, k−1 1 1 l−1 l−1 l+1 l+1 m m l Closed sets of functions Page 7 of 12 61 [k] for all h ∈ P with i ∈ [m]\{l}, for all l ≤ m, λ ∈ F , x ∈ F ,and i l q i i l q [k] h ∈ R , since t is 0-absorbing in [n]. Thus we can observe that: l k−1 1 m r (λ e ,...,λ e ) k 1 m 1 1 k k m [k] F F c (h ) q q i 1 m i=1 = (−1) t (h (λ e ),...,h (λ e )) k−1 1 1 m m 1 1 [k] h ∈P k k m [k] F F c (h ) q q i 1 m i=1 = (−1) t (h (λ e ),...,h (λ e )) k−1 1 1 m m 1 1 [k] h ∈T k k F F q q 1 m = t (h (λ e ),...,h (λ e )) k−1 1 1 m m 1 1 [k] h ∈T k−1 k−1 1 m = t (λ e ,...,λ e ) k−1 1 m 1 1 [k] h ∈T k−1 k−1 1 m = q · t (λ e ,...,λ e ) i k−1 1 m 1 1 i=1 k k F F q q 1 m = q · t (λ e ,...,λ e ). i k 1 m 1 1 i=1 Thus r = q · t . k i k i=1 Because of (3.1) and the inductive hypothesis, we have r ∈ Clg({t }) k k−1 m m [1] [1] ⊆ Clg(C ). Thus q t ∈ Clg(C ). Since q = 0 modulo p we have i k i i=1 i=1 [1] that t ∈ Clg(C ) and this concludes the induction proof. Lemma 3.4. Let q ,...,q and p be powers of primes with q and p co- 1 m i i=1 prime and let K = F .Let C be an (F , K)-linearly closed clonoid, let q p i=1 i [1] I ⊆ [m] and let f ∈ C be 0-absorbing in I.Then f ∈ Clg(C ). Proof. Let K = F and let C := {g |∃g ∈ C : g | = g}. By Lemma 1 q 1 K i∈I i 1 [1] [1] 3.2 f ∈ Clg(C ) if and only if f | ∈ Clg(C | ) and we observe that f | K K 1 K 1 is 0-absorbing in I . Thus without loss of generality we fix I =[m]. The strategy is to interpolate f in all the distinct products of lines of the form {(λ b ,...,λ b ) | (λ ,...,λ ) ∈ F , b ∈ F \{(0,..., 0)}. To this 1 1 m m 1 m q i i=1 i q end let R = {L | 1 ≤ j ≤ (q − 1)/(q − 1) = s} be the set of all s distinct j i i=1 i products of lines of F and let l ∈ F be such that (l ,..., l ) q (i,j) (1,j) (m,j) i=1 i q generates the products of m lines L ,for 1 ≤ j ≤ s,1 ≤ i ≤ m. For all 1 ≤ j ≤ s,let f : F → F be defined by: L p j i=1 q f (λ l ,...,λ l )= f (λ l ,...,λ l ) L 1 (1,j) m (m,j) 1 (1,j) m (m,j) m m for (λ ,...,λ ) ∈ F and f (x ) = 0 for all x ∈ F \{(λ l , 1 m q L 1 (1,j) i=1 i j i=1 q ...,λ l ) | (λ ,...,λ ) ∈ F }. m (m,j) 1 m q i=1 i Claim 1: f = f . j=1 j 61 Page 8 of 12 S. Fioravanti Algebra Univers. Since f is 0-absorbing in [m]wehavethat: f (λ l ,...,λ l )= f (λ l ,...,λ l ) L 1 (1,z) m (m,z) L 1 (1,z) m (m,z) j z j=1 = f (λ l ,...,λ l ) 1 (1,z) m (m,z) for all (λ ,...,λ ) ∈ F and z ∈ [s], since for all j ,j ∈ [s], L 1 m q 1 2 j i 1 i=1 and L intersect only in points of the form (x ,..., x ) ∈ F with j 1 m 2 q i=1 i x =(0,..., 0) for some i ∈ [m]. Let 1 ≤ j ≤ s and let g : F → F be a function such that: q p i=1 i f (λ l ,...,λ l )= g(λ ,...,λ )= f (λ l ,...,λ l ) L 1 (1,j) m (m,j) 1 m 1 (1,j) m (m,j) [1] for all (λ ,...,λ ) ∈ F . Then g ∈ C . 1 m q i=1 [1] Claim 2: f ∈ Clg(C ) for all L ∈ R. L j We can observe that f (λ l ,...,λ l )= g(λ ,...,λ ) for all L 1 (1,j) m (m,j) 1 m (λ ,..., λ ) ∈ F ,and f (x ,..., x ) = 0 for all (x ,..., x ) ∈ 1 m q L 1 m 1 m i=1 i j m m F \{(λ l , ...,λ l ) | (λ ,...,λ ) ∈ F }. Furthermore, 1 (1,j) m (m,j) 1 m q q i i=1 i i=1 [1] g is 0-absorbing in [m]. By Lemmata 3.1 and 3.3, f ∈ Clg(C ), which [1] concludes the proof of f ∈ Clg(C ). We are now ready to prove that an (F, K)-linearly closed clonoid C is generated by its unary part. Theorem 3.5. Let q ,...,q and p be powers of primes with q and p 1 m i i=1 coprime and let K = F . Then every (F , K)-linearly closed clonoid C is q p i=1 i [1] generated by its unary functions. Thus C = Clg(C ). Proof. The inclusion ⊇ is obvious. For the other inclusion let C be an (F , K)- linearly closed clonoid and let f be an n-ary function in C. By Lemma 2.1 with A = F and 0 =(0 ,..., 0 ), f can be split in the sum of n-ary functions i A F F q i q q i i i f such that for each I ⊆ [m], f is 0-absorbing in I. Furthermore, each I I I⊆[m] function f lies in the subgroup F of F that is generated by the functions (I) x → f (x ), where I ⊆ [m] and thus each summand f is in C. By Lemma [1] [1] 3.4 each of these summands is in Clg(C ). and thus f ∈ Clg(C ). The next corollary of Theorem 3.5 and the following theorem tell us that there are only finitely many distinct (F, K)-linearly closed clonoids. Corollary 3.6. Let q ,...,q and p be powers of primes with q and p 1 m i i=1 coprime and let K = F .Let C and D be two (F , K)-linearly closed q p i=1 [1] [1] clonoids. Then C = D if and only if C = D . Let us denote by L(F, K) the lattice of all (F, K)-linearly closed clonoids. We define the functions ρ : L(F, K) →L(F , K) such that for all 1 ≤ i ≤ s i p and for all C ∈L(F, K): ρ (C):= {f | there exists g ∈ C : f = π ◦ g}, (3.2) where with π we denote the projection over the i-th component of the product of fields F. Closed sets of functions Page 9 of 12 61 s m Theorem 3.7. Let F = F and K = F be products of finite fields. p q i i i=1 i=1 Then the lattice of all (F, K)-linearly closed clonoids is isomorphic to the direct product of the lattices of all (F , K)-linearly closed clonoids with 1 ≤ i ≤ s. Proof. Let us define the function ρ : L(F, K) → L(F , K) such that i=1 i ρ(C):=(ρ (C),...,ρ (C)). Clearly ρ is well-defined. Conversely, let ψ : L 1 s i=1 (F , K) →L(F, K) be defined by: [k] [k] ψ(C ,...,C )= {f : x → (f (x ),...,f (x )) | f ∈ C ,...,f ∈ C }. 1 s 1 s 1 s 1 s k∈N From this definition it is clear that ψ is well defined. Furthermore, ρψ(C ,...,C )=(C ,...,C ) 1 s 1 s and C ⊆ ψρ(C) for all (C ,...,C ) ∈ L(F , K)and C ∈L(F, K). 1 s p i=1 To prove that C ⊇ ψρ(C)let f ∈ ψρ(C). Then there exists (f ,...,f ) ∈ 1 s ρ(C) such that f = π ◦ f for all i ∈ [s]. By definition of ρ, there exist g ,...,g ∈ C such that f = π ◦ g for all i ∈ [s]. Let a ∈ F be such 1 s i i i that a (j)=0 for j = i and a (i) = 1. It is easy to check that the function i i f = a g = f and thus f ∈ C. i i i=1 Hence ρ is a lattice isomorphism. s m Proof of Theorem 1.2. Let F = F and K = F be products of p q i i i=1 i=1 finite with |K| and |F| coprime. Let C ∈L(F, K). By Theorem 3.7 C is uniquely determined by its projections C = ρ (C),...,C = ρ (C) where ρ is defined 1 1 s s i in (3.2). By Theorem 3.5 we have that for all i ∈ [s]every (F , K)-linearly [1] closed clonoid C is uniquely determined by its unary part C .Thus C is [1] uniquely determined by its unary part C . 4. The lattice of all (F, K)-linearly closed clonoids In this section we characterize the structure of the lattice L(F, K) of all (F, K)- linearly closed clonoids through a description of their unary parts. Let F = s m F and K = F be products of finite fields such that |K| and |F| p q i=1 i j=1 j are coprime numbers. We will see that L(F, K) is isomorphic to the product of the lattices × K × of all F [K ]-submodules of F , where K is the multiplicative monoid of i p K = F . In order to characterize the lattice of all (F, K)-linearly closed i=1 i clonoids we need the definition of monoid ring. Definition 4.1. Let M, · be a commutative monoid and let R, +,  be a commutative ring with identity. Let S := {f ∈ R | f (a) = 0 for only finitely many a ∈ M }. We define the monoid ring of M over R as the ring (S, +, ·), where + is the point-wise addition of functions and the multiplication is defined as f · g : R → M which maps each m ∈ M into: f (m )g(m ). 1 2 m ,m ∈M,m m =m 1 2 1 2 61 Page 10 of 12 S. Fioravanti Algebra Univers. We denote by R[M ] the monoid ring of M over R. Following the notation in [3] for all a ∈ A we define τ to be the element of R with τ (a)=1and a a τ (M \{a})= {0}. We observe that for all f ∈ R[M ] there is an r ∈ R such that f = r τ and that we can multiply such expressions with the rule a a a∈M τ · τ = τ . a b ab Definition 4.2. Let M be a commutative monoid and let R be a commutative ring. We denote by R the R[M ]-module with the action ∗ defined by: (τ ∗ f )(x)= f (ax), for all a ∈ M and f ∈ R . Let K be the multiplicative monoid of K = F . We can observe i=1 i × K K that V is an F [K ]-submodule of F if and only if it is a subspace of F p p satisfying (x ,...,x ) → f (a x ,...,a x ) ∈ V, (4.1) 1 m 1 1 m m for all f ∈ V and (a ,...,a ) ∈ F . Clearly the following lemma holds. 1 m q i=1 i Lemma 4.3. Let p, q ,...q be powers of primes and let K = F .Let 1 m q i=1 V ⊆ F .Then V is the unary part of an (F , K)-linearly closed clonoid if and × K only if is an F [K ]-submodule of F . Together with Theorem 3.7 this immediately yields the following. m s Corollary 4.4. Let K = F and F = F be products of finite fields q p i i i=1 i=1 [1] such that |K| and |F| are coprime. Then the function π that sends an (F, K)- linearly closed clonoid to its unary part is an isomorphism between the lattice of all (F, K)-linearly closed clonoids and the direct product of the lattices of all × K F [K ]-submodules of F . i p With the same strategy of [6, Lemma 5.6] we obtain the following Lemma. m s Lemma 4.5. Let K = F and F = F be products of finite fields q p i=1 i i=1 i such that |K| and |F| are coprime. Then every (F, K)-linearly closed clonoid is finitely related. The next step is to characterize the lattice of all F [K ]-submodules of K × K F . To this end we observe that V is an F [K ]-submodule of F if and only p p if is a subspace of F satisfying (4.1). We can provide a bound for the lattice of all (F, K)-linearly closed clonoids given by the number of subspaces of F . Remark 4.6. It is a well-known fact in linear algebra that the number of k- dimensional subspaces of an n-dimensional vector space V over a finite field F is the Gaussian binomial coefficient: n−k+i n q − 1 = . (4.2) k q − 1 i=1 Closed sets of functions Page 11 of 12 61 From this remark we directly obtain the bound of Theorem 1.3. In order to determine the exact cardinality of the lattice of all (F, K)-linearly closed clonoids we have to deal with the problem to find the F [K ]-submodules of F . We will not study this problem here because we think that this is an interesting problem that deserves an own research. Acknowledgements The author thanks Erhard Aichinger, who inspired this paper, Erkko Lehtonen who reviewed my Ph.D. thesis, and Sebastian Kreinecker for many hours of fruitful discussions. The author thanks the referees for their useful suggestions. Funding Open access funding provided by Johannes Kepler University Linz. Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and re- production in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regu- lation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons. org/licenses/by/4.0/. Publisher’s Note Springer Nature remains neutral with regard to jurisdic- tional claims in published maps and institutional affiliations. References [1] Aichinger, E.: Solving systems of equations in supernilpotent algebras. In: 44th International Symposium on Mathematical Foundations of Computer Science, LIPIcs. Leibniz Int. Proc. Inform, 138, Art. No. 72, 9, Schloss Dagstuhl. Leibniz- Zent. Inform., Wadern (2019) [2] Aichinger, E., Mayr, P.: Finitely generated equational classes. J. Pure Appl. Algebra 220(8), 2816–2827 (2016) [3] Aichinger, E., Moosbauer, J.: Chevalley-Warning type results on abelian groups. J. 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Algebra Universalis 80(4), 53 (2019) [15] Szendrei, A.: Clones in universal algebra. In: S´ eminaire de Math´ ematiques Sup´ erieures. Seminar on Higher Mathematics, vol. 99. Presses de l’Universit´ e de Montr´ eal, Montreal (1986) Stefano Fioravanti Institut fur ¨ Algebra Johannes Kepler Universit¨ at Linz 4040 Linz Austria e-mail: stefano.fioravanti66@gmail.com Received: 4 September 2020. Accepted: 11 August 2021.

Journal

algebra universalis – Springer Journals

Published: Nov 1, 2021

Keywords: Clonoids; Clones; 08A40

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