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2-affine complete algebras need not be affine complete Erhard Aichinger Abstract. For each k ∈ N, we exhibit a finite algebra Rk such that Rk is k-affine complete, but not (k + 1)-affine complete; this means that every k-ary congruence preserving function on Rk lies in Polk Rk , but there is a (k + 1)-ary congruence preserving function of Rk that does not lie in Polk+1 Rk .
1. Introduction We use the notation of [11]. We abbreviate a vector a1 , a2 , . . . , ak by a. For a congruence α of an algebra A = A; F and for a, b ∈ Ak , we write a ≡ b (mod α) if ai , bi ∈ α for all i = 1, 2, . . . , k. We write a/α for {b ∈ Ak | a ≡ b (mod α)}. By ΘA (a, b) we denote the smallest congruence of A that contains {a1 , b1 , a2 , b2 , . . . , ak , bk }. A function c : Ak → A is congruence preserving if we have c(a), c(b) ∈ ΘA (a, b) for all a, b ∈ Ak . Equivalently, a function c : Ak → A is congruence preserving iff Con A; F = Con A; F ∪ {c}. Congruence preserving functions have also been called congruence compatible operations in [9]. By Compk A, we denote the set of all k-ary congruence preserving functions on A. The set of all k-ary polynomial functions on A will be written as Polk A (cf. [11, Definition 4.4], [10]), and the set of all k-ary term functions on A will be denoted by Clok A [11, Definition 4.2]. For every algebra A and for every k ∈ N, we have Polk A ⊆ Compk A. We will now fix a k ∈ N and study algebras in which Polk A = Compk A holds. For k ∈ N, we say that an algebra A is k-affine complete iff every congruence preserving function from Ak to A is a polynomial function, i.e., iff Compk A = Polk A. The algebra A is called affine complete iff it is k-affine complete for all k ∈ N. The notion of affine completeness was introduced in [16] and has been studied further, e.g., in [13, 5, 14]. As the example of the cyclic group of order 2 shows, a 1-affine complete algebra need not be 2-affine complete [12]. However, for many classes of algebras, it has Presented by K. Kaarli. Received September 27, 2001; accepted in final form February 9, 2002. 2000 Mathematics Subject Classification: 08A40. Key words and phrases: Affine completeness, polynomials, congurence preserving functions. 425
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been proved that 2-affine completeness implies affine completeness (cf. [9]). Here, we mention only three classes of algebras for which this implication has been proved: • simple algebras [15], [10, p.35, Theorem 11.2], • abelian algebras in a congruence modular variety [6], [9, Lemma 5.2.3], • finite algebras that lie in a congruence permutable variety and have a distributive congruence lattice [8], [1, Corollary 5.3 (2)]. It has been open whether every 2-affine complete algebra is affine complete. In [9, Problem 3.4.15], it is asked whether there is a natural number k such that every finite k-affine complete algebra is affine complete. In section 2, we use [2, Theorem 2] to show that there is no such k. In section 3, we exhibit an algebra Rk for each k ∈ N such that Rk is k-affine complete, but not (k + 1)-affine complete. For the case k = 2, this algebra is a commutative ring (without unit). In the proof that the algebra Rk is not (k + 1)-affine complete, we will not make use of the results in [2]. 2. A k-affine complete algebra that is not affine complete Let B be the group Z4 × Z2 ; +, −, 0. For every k ∈ N, we define the algebra Bk by Bk := B; Compk B. Since the congruences of an algebra are already determined by its unary congruence preserving functions (in fact, even by its unary polynomial functions), we have Con Bk = Con B for all k ∈ N.
(2.1)
The lattice Con B is isomorphic to the lattice M3,3 in [11, p.39]; it has been drawn in (2.2). In the following lemma, we find that every algebra that lies in a congruence permutable variety and has M3,3 as its congruence lattice must be nilpotent. Lemma 2.1. Let A be an algebra in a congruence permutable variety such that Con A is isomorphic to M3,3 , and let [ · , · ] denote the term-condition commutator operation on Con A as defined in [11, Definition 4.150]. Then we have [[1A , 1A ], 1A ] = 0A . Proof. For α, β ∈ Con A with α ≤ β, we define the centralizer (α : β) of β modulo α as the largest element γ in Con A with [γ, β] ≤ α. Let δ be the join of all atoms of Con A, and let be the meet of all coatoms of Con A. Since, by [11, Lemma 4.154], A/ is abelian, we have [1A , 1A ] ≤ , and hence (δ : 1A ) = 1A . In the lattice Con A, the interval I[δ, 1A ] is projective to the interval I[0A , ]. From the properties of the commutator operation in a congruence permutable variety ([α, β] ≤ α ∧ β, [α, β] ≤ [α, γ] if β ≤ γ, [α, β ∨ γ] = [α, β] ∨ [α, γ] for all α, β, γ ∈ Con A), one can
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infer that projective intervals give rise to the same centralizers (cf. [4, chapter 9, exercise 4]). Thus (0A : ) = (δ : 1A ), and hence (0A : ) = 1A . From this, we obtain [[1A , 1A ], 1A ] ≤ [, 1A ] = 0A . From Lemma 2.1 and (2.1), we obtain that for each k ∈ N, the algebra Bk is nilpotent. Now we count the n-ary congruence preserving functions on B. Let δ be the join of all atoms of Con B, and let be the meet of all coatoms of Con B. Then every congruence ξ of B satisfies ξ ≤ δ or ξ ≥ . This property will be used to construct many congruence preserving functions. f @ f @f
δf @ f f @f @ @f Con B ∼ = M3,3
(2.2)
Lemma 2.2. Let A be a finite algebra that has two congruences , δ with = 0A , δ = 1A such that for all ξ ∈ Con A, ξ ≤ δ or ξ ≥ . (2.3) Then for every n ∈ N, we have: n
|Compn A| ≥ 22 . Proof. We define m := |A/δ|, and we let n be a natural number. Let f be a function from An to A that is constant on each δ-block of An , and has its range contained in one -block. Then (2.3) yields that f is congruence preserving. Since there are mn δ-blocks, and since by = 0A , at least one -block contains at least two elements of n A, there are at least 2m such functions. More formally, we proceed as follows: We choose a1 , a2 , . . . , am ∈ A such that a1 /δ ∪ a2 /δ ∪ · · · ∪ am /δ = A, and we choose c, d ∈ A such that c = d and c, d ∈ . For every subset S of {a1 , a2 , . . . , am }n , we define a function fS : An → A by d if there is s ∈ S with s ≡ x (mod δ) , fS (x) := c otherwise. Every fS is congruence preserving. To show this, we fix x, y ∈ An . If x ≡ y (mod δ), then fS (x) = fS (y), and thus fS (x), fS (y) ∈ ΘA (x, y). If x ≡ y (mod δ), then ΘA (x, y) ≤ δ, and hence by (2.3), we have ΘA (x, y) ≥ . Since fS maps everything into one -block, we have fS (x), fS (y) ∈ ΘA (x, y) also in this case. Thus fS lies in Compn A. Since S = x ∈ {a1 , a2 , . . . , am }n | fS (x) = d ,
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the mapping S → fS is injective, and thus we have constructed 2m different n-ary congruence preserving functions on A. As a consequence, we see that for the group B, we have n
|Compn B| ≥ 22 for all n ∈ N. The result [2, Theorem 2] gives an upper bound on the number of n-ary polynomial functions for certain nilpotent algebras. Using this result, we will obtain that some algebras cannot be affine complete. Lemma 2.3. Let A be a finite algebra in a congruence modular variety of finite type. We assume that A is nilpotent and furthermore a direct product of algebras of prime power order. In addition, we assume that A has two congruences , δ with = 0A , δ = 1A such that for all ξ ∈ Con A we have ξ ≤ δ or ξ ≥ . Then A is not affine complete. Proof. Let m := |A|. Seeking a contradiction, we suppose that A is n-affine complete for all n ∈ N. From this and Lemma 2.2, we obtain n
|Poln A| = |Compn A| ≥ 22 for all n ∈ N. Let {a1 , a2 , . . . , am } be the universe of A. Then, every n-ary polynomial function p(x1 , x2 , . . . , xn ) can be written as t(a1 , a2 , . . . , am , x1 , x2 , . . . , xn ), where t is an (m + n)-ary term function on A. Hence, we have |Poln A| ≤ |Clom+n A| for all n ∈ N. From this, we obtain
r−m
|Clor A| ≥ 22
for all r > m.
(2.4)
We note that |Clor A| is equal to the number of elements of the free algebra with r generators in the variety generated by A ([3, p. 144, section VI.7, Corollary 3]). Thus by [2, Theorem 2], there is a polynomial p such that for all r ∈ N, we have |Clor A| ≤ 2p(r) , which contradicts (2.4). Theorem 2.4. Let k ∈ N. The algebra Bk is k-affine complete, but not affine complete. Proof. In order to show that Bk is k-affine complete, we observe that every kary congruence preserving function of Bk is a fundamental operation of Bk , and therefore a polynomial function. Now we show that Bk is not affine complete. For k = 1, we observe that B1 has only unary fundamental operations. Now f : B × B → B, x, y → x + y, is congruence preserving. The function f depends on both variables and can therefore not be a polynomial function of B1 . Hence B1 is not 2-affine complete. Now we assume k ≥ 2. Then Bk has the group B as a reduct, and thus the variety generated by Bk is congruence permutable and
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therefore congruence modular. Thus, Bk satisfies the assumptions of Lemma 2.3, and therefore it is not affine complete. 3. A commutative ring that is 2-affine complete and not 3-affine complete, and algebras that are k-affine complete, but not (k + 1)-affine complete Let R := Z4 × Z2 . For each k ∈ N, we define an algebra Rk by Rk := R; +, fk , where fk is the k-ary operation on R given by fk ([ wv11 ] , [ wv22 ] , . . . , [ wvkk ]) :=
2·v
1 ·v2 · ··· ·vk
0
for all v1 , v2 , . . . , vk ∈ Z4 and w1 , w2 , . . . , wk ∈ Z2 . Here 2·v1 ·v2 · · · · ·vk is computed using the usual multiplication on Z4 . We observe that for each i ∈ {1, 2, . . . , k}, we have fk (x1 , x2 , . . . , xi−1 , xi + y, xi+1 , . . . , xk ) = fk (x1 , x2 , . . . , xi−1 , xi , xi+1 , . . . , xk ) + fk (x1 , x2 , . . . , xi−1 , y, xi+1 , . . . , xk ) for all x1 , x2 , . . . , xk , y ∈ Z4 × Z2 ,
(3.1)
and fk (x1 , x2 , . . . , xi−1 , fk (y1 , y2 , . . . , yk ), xi+1 , . . . , xk ) = 0 for all x, y ∈ (Z4 × Z2 )k .
(3.2)
Proposition 3.1. Let k ∈ N. Then Rk is k-affine complete. Proof. We fix k ∈ N. First, we prove Con Z4 × Z2 ; +, fk = Con Z4 × Z2 ; +.
(3.3)
To prove this, we show that fk is a congruence preserving k-ary function on Z4 × Z2 ; +. Let δ be the congruence of Z4 × Z2 ; + generated by [ 00 ] , [ 01 ] and [ 00 ] , [ 20 ], and let be the congruence of Z4 × Z2 ; + that is generated by [ 00 ] , [ 20 ]. Now we observe that fk satisfies 2 [ 0 ] if xi , [ 10 ] ∈ δ for all i ∈ {1, 2, . . . , k}, (3.4) fk (x1 , x2 , . . . , xk ) = [ 00 ] otherwise. Hence fk is constant on each δ-class and its range is contained in one -class. Since every congruence ξ of Z4 × Z2 ; + satisfies ξ ≤ δ or ξ ≥ , we obtain (by the same argument as the one used in the proof of Lemma 2.2 for showing that fS is congruence preserving) that fk is congruence preserving. This proves (3.3).
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Now we show that Rk is k-affine complete. Let c be a congruence preserving function from Rk to R. The quotient Rk / is isomorphic to the algebra Z2 × Z2 ; +, z, where z(x) := [ 00 ] for all x ∈ (Z2 × Z2 )k . This algebra is polynomially equivalent to the group Z2 × Z2 ; +, −, 0. By [12], this group is k-affine complete. Hence there is a polynomial function p ∈ Polk (Rk /) such that p(x/) = c(x)/ for all x ∈ Rk . Lifting the coefficients of p, we obtain a polynomial function p ∈ Polk Rk such that p (x) ≡ c(x) (mod ) for all x ∈ Rk . The function q : Rk → R, q(x) := c(x)−p (x) is congruence preserving, and satisfies q(Rk ) ⊆ [ 00 ] /. ˆ : (A × A)k → For every k-ary function h on an algebra A, we define a function h A × A by ˆ 1 , b1 , a2 , b2 , . . . , ak , bk ) := h(a1 , a2 , . . . , ak ), h(b1 , b2 , . . . , bk ). h(a If α, β, σ, τ are congruences of A such that the intervals I[α, β] and I[σ, τ ] are projective, and if h is a congruence preserving k-ary function on A, then we have ˆ k ) ⊆ α. ˆ k ) ⊆ σ if and only if h(β h(τ
(3.5)
We know that the function q satisfies qˆ((1R )k ) ⊆ . Let µ be an atom of Con Rk with µ = . Since the interval I[δ, 1R ] is projective to the intervals I[, δ] and I[µ, δ] in Con Rk , the property (3.5) yields qˆ(δ k ) ⊆ ∧µ = 0A . This means that q satisfies the following property: For all a, b ∈ Rk with a ≡ b (mod δ) we have q(a) = q(b).
(3.6)
k
Hence, we see that q is one of the 22 functions from Rk to R that satisfy (3.6) and map Rk into [ 00 ] /. For each a ∈ {[ 00 ] , [ 10 ]}k , we define a function ga ∈ Polk Rk by ga (x) = fk x − a + [ 10 ] , [ 10 ] , . . . , [ 10 ] for all x ∈ Rk . Then from (3.4), we obtain that ga (x) is equal to [ 20 ] if x ≡ a (mod δ), and equal to [ 00 ], otherwise. Let S(q) be the set defined by S(q) := a ∈ {[ 00 ] , [ 10 ]}k q(a) = [ 20 ] . Using (3.6), we obtain q(x) =
ga (x) for all x ∈ Rk .
a∈S(q)
Hence we have q ∈ Polk Rk , and therefore c ∈ Polk Rk .
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From Proposition 3.1, we also obtain that for k ≥ 2, the algebra Rk is polynomially equivalent to the algebra Bk defined in section 2. In order to show that Rk is not (k + 1)-affine complete, we will need an operator on the (k + 1)-ary functions on R. Definition 3.2. Let A; +, −, 0 be an abelian group, let n ∈ N, and let p : An → A be a function. We will define an (n+ 1)-ary function D(n) (p) : An+1 → A as follows: Let In := {0, 1} × {0, 2} × · · · × {0, n}. For i1 , i2 , . . . , in ∈ In , we define w(i1 , i2 , . . . , in ) as the number of nonzeros in i1 , i2 , . . . , in , i.e., as |{l ∈ {1, 2, . . . , n} | il = 0}|. Now we define the function D(n) (p) : An+1 → A by D(n) (p) (x0 , x1 , x2 , . . . , xn ) := (−1)w(i1 ,i2 ,...,in ) p(xi1 , xi2 , . . . , xin ). i1 ,i2 ,...,in ∈In
Hence, for p : A1 → A, we have D(1) (p) (x0 , x1 ) = p(x0 )−p(x1 ), and for p : A2 → A, we have D(2) (p) (x0 , x1 , x2 ) = p(x0 , x0 ) − p(x0 , x2 ) − p(x1 , x0 ) + p(x1 , x2 ). We will use the following property of the operator D(n) . Lemma 3.3. Let n ∈ N, let A be an abelian group, and let p, q : An → A. Then we have D(n) (p + q) = D(n) (p) + D(n) (q). Furthermore, if there exists a number j ∈ {1, 2, . . . , n} such that p(x1 , x2 , . . . , xj−1 , a, xj+1 , . . . , xn ) = p(x1 , x2 , . . . , xj−1 , b, xj+1 , . . . , xn ) for all x ∈ An , a, b ∈ A, then D(n) (p) = 0. Proof. The linearity of D(n) is obvious. For the second property, we observe that, if p does not depend on its j-th argument, then, in the sum that defines D(n) (p) (x), the summand p(xi1 , xi2 , . . . , xij−1 , xj , xij+1 , . . . xin ) cancels with the summand p(xi1 , xi2 , . . . , xij−1 , x0 , xij+1 , . . . xin ). Theorem 3.4. Let k ∈ N. Then Rk is not (k + 1)-affine complete. Proof. We fix k ∈ N. For each n ∈ N, we define a subset Bk,n of Clon Rk as follows: the elements of Bk,n are the projection functions x1 , x2 , . . . , xn → xi , where i ∈ {1, 2, . . . , n}, and all functions x1 , x2 , . . . , xn → fk (xi1 , xi2 , . . . , xik ), where i1 , i2 , . . . , ik ∈ {1, 2, . . . , n}k . Now we prove: Every function in Clon Rk is a sum of functions that lie in Bk,n .
(3.7)
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Formally, (3.7) states that the subuniverse of Clon Rk ; + generated by Bk,n is equal to Clon Rk . To prove (3.7), we show that for every term t that is built from x1 , x2 , . . . , xn with the operation symbols + (binary) and fk (k-ary), the n-ary function tRk that t induces on Rk is a sum of functions from Bk,n . We proceed by induction on the length of t. • If t = xi for some i ∈ {1, 2, . . . , n}, then tRk lies in Bk,n . Rk k • If t = s1+ s2 , both sR 1 and s2 are sums of functions from Bk,n by the induction Rk hypothesis, and hence t is a sum of functions from Bk,n . • Now we consider the case that there are terms s1 , s2 , . . . , sk such that t = fk (s1 , s2 , . . . , sk ). If each sj is a variable xij , then we have tRk ∈ Bk,n by the definition of Bk,n . Now we assume that j is such that sj is not a variable. If sj = s+ s , then (3.1) yields that t = fk (s1 , s2 , . . . , sj−1 , s+ s , sj+1 , . . . , sk ) induces the same function on Rk as the term t+t , where t = fk (s1 , s2 , . . . , sj−1 , s , sj+1 , . . . , sk ), t = fk (s1 , s2 , . . . , sj−1 , s , sj+1 , . . . , sk ). Both t and t are shorter than t, and hence, by the induction hypothesis, t and t induce functions on Rk that are sums of functions from Bk,n . Therefore, the function induced by t is a sum of functions from Bk,n , as well. If there are terms r1 , r2 , . . . , rk such that sj = fk (r1 , r2 , . . . , rk ), then (3.2) yields that t induces the constant function that maps all elements of Rn to 0. This completes the proof of (3.7). Now we prove that every p ∈ Polk+1 Rk satisfies D(k+1) (p) (x) = 0 for all x ∈ Rk+2 .
(3.8)
We fix a polynomial function p ∈ Polk+1 Rk , and we let r1 , r2 , . . . , r8 be an enumeration of the elements of R. We can find a term function t ∈ Clok+9 Rk such that p(x) = t(r1 , r2 , . . . , r8 , x1 , x2 , . . . , xk+1 ) for all x ∈ Rk+1 . The term function t is a sum of some functions that lie in Bk,k+9 . Every function in Bk,k+9 depends on at most k variables. Hence, also p is the sum of functions such that each of the summands depends on at most k variables. Thus, by Lemma 3.3, we have D(k+1) (p) = 0, which proves (3.8). The function fk+1 is a (k + 1)-ary function that lies in Compk+1 Rk . We will now show fk+1 ∈ Polk+1 Rk .
(3.9)
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From a description of fk+1 as in (3.4), we obtain D(k+1) (fk+1 ) ([ 00 ] , [ 10 ] , [ 10 ] , . . . , [ 10 ]) = [ 20 ] .
k+1 times
(p) = 0 for all p ∈ Polk+1 Rk , we obtain that fk+1 is not a Since by (3.8), D polynomial function of Rk , which concludes the proof of (3.9). Hence Rk is not (k + 1)-affine complete. (k+1)
In concluding, we observe that R1 is polynomially equivalent to the group Z4 × Z2 ; +, −, 0, and R2 is a commutative ring without unit. Theorem 3.4 also sharpens Theorem 2.4: for each k ∈ N with k ≥ 2, Rk is polynomially equivalent to Bk . Hence, for k ≥ 2, we obtain from Theorem 3.4 that the algebra Bk is not (k + 1)-affine complete. Acknowledgements. The use of the algebras Rk in section 3 follows a suggestion of P. M. Idziak (cf. also [7]). Several discussions on the topic treated in this paper took place at the workshop “A course in tame congruence theory”, which was held at Budapest in July 2001, and supported by the Paul Erd˝ os Summer Research Center of Mathematics. References 1. E. Aichinger, On Hagemann’s and Herrmann’s characterization of strictly affine complete algebras, Algebra Universalis 44 (2000), 105–121. 2. J. Berman and W. J. Blok, Free spectra of nilpotent varieties, Algebra Universalis 24 (1987), no. 3, 279–282. 3. G. Birkhoff, Lattice theory, American Mathematical Society, Providence, R. I., 1967. 4. R. Freese and R. N. McKenzie, Commutator theory for congruence modular varieties, London Math. Soc. Lecture Note Ser., vol. 125, Cambridge University Press, 1987. 5. J. Hagemann and C. Herrmann, Arithmetical locally equational classes and representation of partial functions, Universal Algebra, Esztergom (Hungary), vol. 29, Colloq. Math. Soc. J´ anos Bolyai, 1982, pp. 345–360. 6. C. Herrmann, Affine algebras in congruence modular varieties, Acta Sci. Math. (Szeged) 41 (1979), no. 1-2, 119–125. 7. P. M. Idziak, Clones containing Mal’tsev operations, Internat. J. Algebra Comput. 9 (1999), no. 2, 213–226. 8. K. Kaarli, Compatible function extension property, Algebra Universalis 17 (1983), 200–207. 9. K. Kaarli and A. F. Pixley, Polynomial completeness in algebraic systems, Chapman & Hall/CRC, Boca Raton, Florida, 2001. 10. H. Lausch and W. N¨ obauer, Algebra of polynomials, North-Holland, Amsterdam, London; American Elsevier Publishing Company, New York, 1973. 11. R. N. McKenzie, G. F. McNulty, and W. F. Taylor, Algebras, lattices, varieties, volume I, Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, California, 1987. ¨ 12. W. N¨ obauer, Uber die affin vollst¨ andigen, endlich erzeugbaren Moduln, Monatshefte f¨ ur Mathematik 82 (1976), 187–198. 13. A. F. Pixley, Characterizations of arithmetical varieties, Algebra Universalis 9 (1979), no. 1, 87–98.
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14. A. F. Pixley, Functional and affine completeness and arithmetical varieties, Proceedings of the NATO Advanced Study Institute and Seminaire de mathematiques superieures, Montreal, Canada, 1991, NATO ASI Ser., Ser. C, Math. Phys. Sci., vol. 389, Kluwer Academic Publishers, 1993, pp. 317–357. 15. W. Sierpi´ nski, Sur les fonctions de plusieurs variables, Fund. Math. 33 (1945), 169–173. 16. H. Werner, Produkte von Kongruenzklassengeometrien universeller Algebren, Math. Z. 121 (1971), 111–140. Erhard Aichinger Institut f¨ ur Algebra, Stochastik, und wissensbasierte mathematische Systeme, Johannes Kepler Universit¨ at Linz, 4040 Linz, Austria e-mail :
[email protected] To access this journal online: http://www.birkhauser.ch
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