Finite aﬃne algebras are fully dualizable

ABSTRACT We show that every finite aﬃne algebra A admits a full duality. In the process, we prove that A also allows a strong duality, and that the duality may be induced by a dualizing structure of finite type. We give an explicit bound on the arities of the partial and total operations appearing in . In addition, we show that the enriched partial hom-clone of A is finitely generated as a clone.


Introduction
A full duality represents elements of abstract algebraic structures by using functions on a topological space that is o en enriched with a relational and/or operational structure, and vice versa. This representation allows us to solve algebraic questions by the way of additional structure. For example, in Stone duality, Boolean algebras are dual to Boolean spaces. Under this correspondence, the familiar Cantor space is dual to the denumerable free Boolean algebra, with many of the universal properties of the Cantor space being dual counterparts to the natural universal properties of being a free algebra (the universal mapping property for example).
In a natural full duality, the representation is constructed in a certain systematic way, using a generating algebra A and a corresponding topological structure A ∼ , called an "alter ego" of A. We say that A is fully dualizable, if there exists an alter ego A ∼ such that every algebra from the quasivariety generated by A and every topological structure from the topological quasivariety generated by A ∼ has a representation. We remark that in case of a full duality, the correspondence can be extended to homomorphisms and continuous structure preserving maps, yielding a category-theoretic dual equivalence between the corresponding categories.
A full duality is the symmetrized concept of a duality. The de nitions of duality and dualizability di er from that of full duality and full dualizability by requiring that only the algebras in the quasivariety generated by A have duals, while the topological quasivariety generated by A ∼ might contain structures without a representation.
Despite a growing understanding of duality theory, dualizability and full dualizability of an algebra continue to be mysterious properties. For some classes of algebras (such as algebras generating congruence-distributive varieties), there exists a well-behaved dividing line between the dualizable and non-dualizable algebras. In other cases, the partial results available seem to defy any discernable pattern. This latter case includes classes of algebras that are otherwise considered to be well understood, such as Maltsev algebras (or even extensions of groups).
Abelian algebras in congruence-modular varieties are among the most well-behaved classes of algebras, being polynomially equivalent to modules, hence such Abelian algebras are a ne algebras. Surprisingly, most results concerning their dualizability have been announced relatively recently.
In 1995, Davey and Quackenbush [3] showed the dualizability of nite semi-simple Abelian algebras from congruence-modular varieties. Recently, Kearnes and Szendrei [6] established a su cient condition for dualizability, implying in particular the dualizability of nite a ne algebras. Independently, Gillibert also proved the dualizability of nite a ne algebras [5], answering a question from [1].
In this article, we will complete the remaining dualizability question for nite a ne algebras by showing the following Theorem. Theorem 1.1. All nite a ne algebras are fully dualizable.
In fact we will show slightly more. First, we show that a full duality can be obtained by an alter ego A ∼ of nite type, and we give an explicit bound on the arity of the (partial) functions and relations in A ∼ . Second, we establish full dualizability by showing that every nite a ne algebra satis es the stronger property of (adequately named) strong dualizability.
Additionally, we obtain a structural result in clone theory. An n-ary partial operation f over A is compatible with an algebra structure A if the domain D of f is a subalgebra of A n , and f is a homomorphism from D to A. We show that the clone of all partial functions compatible with an a ne algebra A is nitely generated as a clone (Corollary 5.4).
The proof of our main theorem relies on a technical condition from [9] (Theorem 2.7), that requires us to nd a suitable factorization for each partial A-compatible function through a bounded set of partial functions. Our article is structured around this requirement as follows: In Section 2 we de ne basic terms and establish several results about a ne algebras. Section 3 provides a technical result about the factorization of projections on partial domains in the quasivariety generated by A. This result will allow us to concentrate our further considerations on partial homomorphisms without proper extensions. In Section 4, we prove a crucial theorem about those partial homomorphisms: namely, a partial homomorphism that cannot be extended must have a large domain. This result is then used in Section 5 to prove a factorization property for all partial homomorphisms, and to prove our main theorem.
Sections 6 and 7 contain an example calculation and a list of problems motivated by our research. Moreover, we have included an appendix that gives explicit bounds on the number of various algebraic objects. While the results of the appendix are used in our arguments, they are only necessary in establishing an explicit bound on the arities used in a fully dualizing alter ego. A reader without an interest in such an explicit bound may ignore the appendix and instead check the simple fact that all quantities in our argument are nite.

Basic concepts
Given an algebra A, we denote by A its underlying set, and by Sub(A) the set of subalgebras of A. The variety Var(A) (respectively, the quasivariety QVar(A)) generated by A is the smallest classes of algebras, with the signature of A that is closed under taking products, subalgebras, and homomorphic images (respectively, products, subalgebras, and isomorphic algebras).
For an arbitrary set X and a variety V, we denote by F V (X) the algebra freely generated by X in V. A subproduct algebra A ≤ i A i is called a subdirect product if π i (A) = A i for each projection π i . An algebra is subdirectly irreducible if whenever it is isomorphic to a subdirect product, it is already isomorphic to one of its factors.
An algebra A is a ne if there exists an Abelian group structure A; +, 0, − such that t(x, y, z) = x − y + z is both a term function of A and a homomorphism from A 3 to A. Such a term t is called a Maltsev term. A class of algebras C is a ne if all of its algebras are. In the case of an a ne variety V, it is easy to see that we may choose one term t that witnesses the a nity simultaneously for all members of V (e.g. we could take the term witnessing the a nity of F V (ω)).
For example, let G be an Abelian group, and consider t : G 3 → G, (x, y, z) → x − y + z. Then (G, t) is an a ne algebra. Or let A be an a ne space over some eld F. For all λ 1 , . . . , λ n in F such that λ 1 + · · · + λ n = 1, consider the operation f (λ 1 ,...,λ n ) : A n → A which maps (a 1 , . . . , a n ) to the barycenter of (a 1 , . . . , a n ) with weight (λ 1 , . . . , λ n ). Denote by P the set of all such operations. Then (A, P) is an a ne algebra.
The notion of an a ne algebra is closely related to that of an Abelian algebra. An algebra A is Abelian if [1 A , 1 A ] = 0 A , where 1 A and 0 A are the universal and trivial relations on A, and [·, ·] denotes the binary commutator on the congruences of A (we refer to [4] for the de nition of the commutator). In congruence modular varieties, Abelian and a ne algebras coincide [4,Corollary 5.9].
We repeat several results about congruences of a ne algebras from [5].
De nition 2.1 ([5], De nition 3.1). Let A be an a ne algebra and B ∈ Sub(A). The congruence generated by B, denoted by B , is the smallest congruence of A containing B 2 .
We remark that not every congruence of A can be written in the form B for some subalgebra B of A, and that we might have B = C with B = C.

Lemma 2.2 ([5], Lemma 3.3). Let
A be an a ne algebra, let B ∈ Sub(A), and let t be a term witnessing the a nity of A. Then Note that this result implies that B is a congruence class of B . It is well known that any a ne variety is polynomially equivalent to a variety of modules. Moreover, if a variety is generated by a nite a ne algebra A with Maltsev term t, and u ∈ A, then the corresponding ring is isomorphic to the set of all unary polynomial functions of A xing u, with addition de ned by f + g = t(f , u, g), and multiplication as composition of maps. In this case, the zero of the ring is the constant map with image u, and the unit is the identity map. Also note that any polynomial function xing u can be chosen as x → f (x, u), for some idempotent binary term f of A.
Lemma 2.4. Let V be a variety generated by an a ne algebra A, where |A| = p α 1 1 · · · p α k k for distinct primes p 1 , . . . , p k . Let R be a ring associated to V. Then |R| divides p Proof. Let u ∈ A. We can assume that R is the set all unary polynomial functions of A xing u, as de ned before. Consider the group operation + over A de ned by x + y = t(x, u, y) for all x, y ∈ A. Note that by construction R; + is a subgroup of A; + A . As A is an a ne algebra it follows that each f ∈ R is compatible with t, hence for all x, y ∈ A the following equalities hold Therefore R; + is a subgroup of Hom( A; + , A; + ); + . By Lemma 8.1(2), |R| divides Given sets A, B we denote by F(A, B) the set of all maps A → B. Let A be a set. Given n ∈ N we consider the set of n-ary partial operations de ned by: The set of all partial operations over A is Note that alternative de nitions distinguish empty functions of di erent arity; the di erence is immaterial for our results. Denote by π n i : A n → A, x → x i the canonical projection for all positive integers n and all 1 ≤ i ≤ n. A partial clone over A is a set F ⊆ C(A), such that F contains all projections and is closed under composition of partial functions.
Let F be a partial clone over A. A domain of arity n of F is a subset D of A n such that there exists f ∈ F with dom f = D.
Lemma 2.5. Let F be a partial clone over a set A, n a positive integer, and C, D domains of arity n of F. The following statements hold.
(1) For all 1 ≤ i ≤ n, the restriction of π n i to D belongs to F. Proof. Take f : C → A and g : D → A in F.
We will consider partial functions whose domains are subalgebras and which are homomorphisms. For algebras A ≤ B and C and a homomorphism f from A to C, we say that f has a proper extension if there is an algebra D with A < D ≤ B and a homomorphism f ′ from D to C that extends f .
Our aim is to establish that all a ne algebras are fully dualizable. We will now recall the de nition of full dualizability and of related terms from [2]. We remark this we will not actually use any of these de nitions, instead relying on established technical results to prove our claims.
For any nite algebra A, consider a topological structure A ∼ = A; F, R, τ on the universe of A, where F is a set of (total or partial) operations, R is a set of relations, and τ is the discrete topology. If each fundamental operation of A preserves every relation in R and is compatible with every function in F, then A ∼ is called an alter ego of A.
To A and A ∼ we attach two categories A, X , respectively. Here A consists of all algebras in QVar(A) together with their homomorphisms. The objects of X are all isomorphic images of (topologically) closed substructures of products of A ∼ , where the products are taken over non-empty index sets. The morphisms of X consist of all continuous mappings that preserve relations of R and are compatible with the operations of F. For B ∈ A, let D(B) ∈ X be the substructure of A ∼ B whose universe consists of all homomorphisms from B to A. Reversely, for X ∈ X , let E(X) ∈ A be the subalgebra of A X whose universe consists of all homomorphisms from X to A ∼ . Then D and E are well de ned and can be extended to contravariant functors between A and X . Now for each B ∈ A, there is a natural embedding e B of B into ED(B) through evaluation. That is, for all b ∈ B, e B (b) maps h to h(b), for all h ∈ D(B). A correspondingly de ned embedding ε X : X → DE(X) exists for all X ∈ X .
We say that an alter ego A ∼ dualizes A, if e B is an isomorphism for each B. If in addition, all ε X are isomorphisms, we say that A ∼

fully dualizes A. An algebra A is [fully] dualizable if there exists an alter ego that [fully] dualizes A.
If A is [fully] dualizable, then D and E induce a dual representation [dual equivalence] between the categories A and X . The aforementioned duality between Boolean algebras and Boolean spaces can be obtained by choosing A as the two-element Boolean algebra and A ∼ as the two-element Boolean space.
Another well-known related concept is that of Pontryagin duality, which can be obtained in a similar fashion by choosing A = A ∼ to be the circle group with its usual topology. This induces a self-duality on the category of all locally compact Abelian groups and continuous homomorphisms. However, Pontryagin duality is not a direct example of our approach, as the circle group is in nite, carries a nondiscrete topology, and both A and A ∼ are topological structures.
As mentioned, we will not be using the dualizability de nitions directly and instead utilize the following results from [2] and [9]. Here, strong dualizability is a special type of full dualizability that we will not de ne, instead referring the reader to [2].
De nition 2.6 ([9]). A nite algebra A has enough total algebraic operations, if there exists ϕ : ω → ω such that for all B ≤ C ≤ A n and every h ∈ hom(B, A), which has an extension to C, there exists Our next result is a special application of the M-shi strong duality Lemma from [2] to the alter ego A, P, τ .

Lemma 2.10 ([2], Lemma 3.2.3).
Let A, P, and τ be as in the Theorem 2.9. Let P ′ ⊆ P be a generating set of P, that is, every h ∈ P is a composition of elements of P ′ and projections. If A, P, τ yields a strong duality on A, then A, P ′ , τ yields a strong duality on A.

A generating set for domains of partial functions
In order to show our main result, we want to establish that every a ne algebra satis es the conditions of De nition 2.6, so that we may use Theorem 2.7. The set X appearing in the de nition can actually be taken as a set of coordinate projections. Hence to establish a necessary bound on X, we need to be able to show that partial compatible functions on A (i.e. homomorphisms from subpowers of A to A), factor though partial compatible functions of bounded arity. As a rst step toward our result, in this section we show that we can generate all possible domains of such functions from a nite set.
The following de nitions are from [5]. Given a ne algebras A and S and a homomorphism k : A → S, let H k (A 2 , S) consist of all homomorphisms f : A 2 → S that satisfy f (x, x) = k(x). We setk ∈ H k (A 2 , S) ask(x, y) = k(y). In [5,Lemma 5.4], it is shown that H k (A 2 , S); +k is an Abelian group (where +k is de ned by f + k g = t(f , k, g) for all f , g in H k (A 2 , S)), and that the isomorphism type of H k (A 2 , S); +k does not depend on k. We let H(A 2 , S); + stand for this isomorphism type.
The following lemma, proved in [5,Lemma 5.7], expresses that a (total) homomorphism f : A n → S can be factored through a small power of A, which does not depend on n but depends only on H(A 2 , S); + . Lemma 3.1. Let A, S be algebras in a variety of a ne algebras. Let L be a positive integer such that H(A 2 , S); + has a family of generators with L − 1 elements. Let f : A n → S be a homomorphism. Then there exists a homomorphism p : A n → A L that is a term in t, and a homomorphism q : . Let A and S be a ne algebras such that |A| = p α 1 1 · · · p α k k and |S| = p , and H(A 2 , S); + has a generating set of size max 1≤i≤k (α i β i ).
Denote by F the partial clone over A generated by t and all π K 1 ↾ C for C a subalgebra of A K . Then π n 1 ↾ D belongs to F for all positive integers n and all subalgebras D of A n .
Proof. Let n be a positive integer and D a completely meet-irreducible subalgebra of A n . Set S = A n / D , and denote by f : A n → S the canonical projection. By Lemma 2.2, {D} is the underlying set of a (one-element) subalgebra of S, moreover, As Therefore, by Lemma 3.1, there exists a homomorphism p : A n → A K , which is a term in t, and a homomorphism q : As {D} is the underlying set of a subalgebra of S and q is a homomorphism, it follows that C is the underlying set of a subalgebra of A K , hence C is a domain of F. Moreover, p is a term of t, thus it follows from Lemma 2.5(4) that p −1 (C) is a domain of F. However, Let B be an arbitrary subalgebra of A n . We can write B as the intersection of nitely many underlying sets of completely subdirectly irreducible subalgebras of A n . Since each of these sets is a domain of F, it follows from Lemma 2.5(2) that B is a domain of F. Therefore, by Lemma 2.5(3), π n 1 ↾ B is in F.

Extensions of Partial homomorphisms
The results of Corollary 3.3 imply that we may generate a partial homomorphism from its extension to a larger domain and a bounded number of partial projections. Thus, the goal of this section is to extend partial homomorphisms of a ne algebras (in a nitely generated variety of a ne algebras). We will show that if the domain of a partial homomorphism is small enough, then the partial homomorphism has a proper extension (cf. Lemma 4.5). We will rst establish this result for modules before generalizing to a ne algebras. .
. It follows that the map is well de ned, and is a homomorphism of modules. Moreover Similarly h extends g.

Lemma 4.2.
Let V be a locally nite variety of algebras, A ∈ V, and n a positive integer. If each nitely generated subalgebra of A is generated by n elements, then A is nite.
Proof. Assuming that A is in nite, there is an in nite sequence (x i ) i∈N of distinct elements of A. Let k = |F V (n)|.
Denote by B the subalgebra of A generated by {x 0 , x 1 , x 2 , . . . , x k }. Note that |B| ≥ k + 1, but B is nitely generated, so is generated by n elements, hence |B| ≤ |F V (n)| = k, a contradiction. Denote by F the R-module freely generated by {u}, so that F ∼ = R as R-modules.
Let B ≤ C in V and f : B → E be a homomorphism. Assume that C/B is not generated by N elements. First note that if all nitely generated submodules of C/B are generated by N elements, then it follows from Lemma 4.2 that C/B is generated by N elements, which contradicts the assumption. Therefore there is a nitely generated submodule Q of C/B, whose minimal number of generators is k ≥ N + 1.
Let P be the submodule of C containing B such that P/B = Q. Note that {x 1 + B, x 2 + B, . . . , x k + B} generates Q if and only if B ∪ {x 1 , . . . , x k } generates P. We say that x 1 , . . . , x k generate P over B.
Given x ∈ C we denote by ϕ x : F → C the unique homomorphism that maps u to x. Note that ϕ x + ϕ y = ϕ x+y for all x, y ∈ C. Pick x 1 , . . . , x k ∈ P\B, generating P over B, such that (ϕ −1 x i (B)) 1≤i≤k is maximal. That is, if y 1 , . . . , y k generate P over B and ϕ −1 The existence of such a sequence follows from the niteness of SubF.
Note that dom f ∩ dom g = {0}, hence it follows from Lemma 4.1 that there exist a homomorphism h : B + ϕ x i (F) → E that extends both f and g. As x i ∈ B, it follows that h is a proper extension of f .
We now assume that S i = {0} for each 1 ≤ i ≤ k. Assume we have i such that S i = F. Hence u, the generator of F, belongs to S i , so Therefore the S i are proper submodules of F, for all 1 ≤ i ≤ n. As k ≥ N + 1 > N and the S i are proper nontrivial submodules of F, it follows that there is a submodule G of F, with G proper and non-trivial, and I ⊆ {1, . . . , k}, such that |I| = k i=1 r i β i and As G is a proper submodule of F, it follows from Lemma 8.1 that |Hom(G, E)| is a proper divisor of p r 1 β 1 1 · · · p r k β k k , hence by Lemma 4.3, there is i ∈ I and a family of integers (u j ) j∈J (where J = I\{i}) such that for all s ∈ G thus ϕ −1 y (B) ⊇ G = S i . It follows from the maximality of (S 1 , . . . , S k ) that ϕ −1 y (B) = S i = G. Set H = ϕ y (F). Let z ∈ H ∩ B, and take s ∈ G such that ϕ y (s) = z. The following equalities hold Note that y = ϕ y (u) ∈ D and y ∈ B, so h is a proper extension of f .
By [8], every locally nite variety V of a ne algebras is generated by a nite algebra A; hence when V is locally nite we can de ne the ring associated to V as the ring associated to A.

Lemma 4.5.
Let V be a locally nite variety of a ne algebras, and R the ring associated to V. Let E ∈ V be nite. Assume that |R| = p r 1 1 · · · p r k k , and |E| = p Proof. Let t be a Maltsev term of V. Note that we may identify the ring corresponding to V with the set R of all idempotent binary terms of V modulo the equational theory of V. Under this correspondence, multiplication is de ned by (λµ)(x, y) = λ(µ(x, y), y), addition is (λ + µ)(x, y) = t(λ(x, y), y, µ(x, y)), the zero is (x, y) → y, and the unit is (x, y) → x.
Let A ∈ V, and for u ∈ A, denote by + u the operation de ned over A by x + u y = t(x, u, y). De ne an action of R over A by λx = λ(x, u) for all x ∈ A, then A; + u , R is a module. Let Assume that C; + v , R /B is generated by x 1 +B, . . . , x N +B. It follows that C/ B is generated by B = v + B, x 1 + B, . . . , x N + B, and so is generated by N + 1 elements; a contradiction. Therefore C; + v , R /B is not generated by N elements. It follows from Lemma 4.4 that as a module homomorphism, f has a proper extension g. We claim that g is also an extension of the V-homomorphism f .
Denote by D the domain of g. Note that C is a reduct of the module structure C; + v , R extended with the constants of V. Moreover, all constants of V are in B ⊆ D, and D is the universe of a submodule of C; + v , R . It follows that D is a subalgebra of C, and that g : D → E is a homomorphism. Hence f has a proper extension.

Factoring partial homomorphisms
The main goal of this section is to factorize a partial homomorphism f : C → E (where C ≤ A n ) through a smaller power D ≤ A N , where N only depends on A and E. This will allow us to use Theorem 2.7 and to prove our main result.
First note that a ne algebras have the congruence extension property.
To be more precise we give the following description of extensions of congruences. (1) For all (x, y) ∈ β, if y ∈ A, then x ∈ A.
Proof. Pick 0 ∈ A and set x+y = t(x, 0, y). It follows that A is stable for +. Also note that x−y = t(x, y, 0) and t(x, y, z) = x − y + z.
We will leave it to the reader to check that β is a congruence that satis es conditions (1) and (2).
Let γ be a congruence of B containing α. Let x, y, a ∈ B such that (x−a, y−a) ∈ α, so (x−a, y−a) ∈ γ , hence (x, y) = (x − a + a, y − a + a) ∈ γ . Therefore γ contains β, hence β is the smallest extension of α to B.
Then for every positive integer n, every subalgebra C of A n , and every homomorphism h : C → E, there exists homomorphisms p : A n → A ℓ and k : p(C) → E such that k • p ↾ C = h. Moreover, we can choose p to be a term in t, the Maltsev term of V.
Proof. Note that most of the homomorphisms and commuting relations used in this proof are illustrated in Figure 1. Set P = |F V (N + 1)| × |E| = p u 1 +e 1 1 · · · p u k +e k k . Let n be a positive integer, C ∈ Sub(A n ), and h : C → E a homomorphism. Let D be a subalgebra of A n , such that h ′ : D → E is a maximal extension of h. Denote by η 1 : C → D the inclusion homomorphism. As h ′ extends h, we have It follows from Lemma 4.5 that A n / D is generated by N + 1 elements, hence |A n / D | divides |F V (N + 1)|. Let ε 1 : D → A n be the inclusion homomorphism. Denote by α the kernel of h ′ and by β the minimal extension of α to A n . Set R = D/α and S = A n /β. Let ε 2 : R → S be the canonical embedding. Let π : D → R and π ′ : A n → S be the canonical projections. Note that As h ′ factors through π , there is an embedding σ : R → E such that Hence |R| = |D/α| divides |E|. It follows from Lemma 5.2 that |S| = |A n /β| = |A n / D | × |D/α| divides P, hence, by Corollary 3.2, the algebra H(A 2 , S); + has a generating family with max 1≤i≤k (α i (e i + u i )) = ℓ − 1 elements. From Lemma 3.1, we have a homomorphism p : A n → A ℓ , which is a term in t, and q : A ℓ → S such that As D is a subalgebra of A n , it follows that p(D) is a subalgebra of A ℓ . Denote by ε 3 : p(D) → A ℓ the canonical embedding. Note that Similarly we denote by η 2 : p(C) → p(D) the inclusion homomorphism, so The following equalities are direct consequences of (5.2), (5.4), and (5.5) So q(ε 3 (p(D))) = ε 2 (π(D)) = ε 2 (R). However, ε 2 is an embedding, so q(ε 3 (p(D))) corresponds to a subalgebra of R; hence, there is a homomorphism u : p(D) → R such that It follows from (5.8) and (5.7) that ε 2 • u • p ↾ D = q • ε 3 • p ↾ D = ε 2 • π . As ε 2 is an embedding it follows that u • p ↾ D = π . (5.9) The following equalities hold = h by (5.1).
Corollary 5.4. Let A be a nite a ne algebra, and p α 1 1 · · · p α k k the prime decomposition of |A|. The enriched partial hom-clone of A is nitely generated by partial operations of arity at most Proof. We may assume that A is non-trivial. Let ℓ be the bound in Theorem 5.3 for the special case E = A. That is, where |F V (N + 1)| = p u 1 1 · · · p u k k , N is as in Lemma 4.5 and V is the variety generated by A. Moreover, let K = 1 + max 1≤i≤k (α 3 i ) (i.e. K is the bound from Corollary 3.3). Given integers 1 ≤ i ≤ n , we denote by π n i : A n → A the canonical projection on the i-th coordinate.
Denote by F the partial clone generated by {t} ∪ X ∪ Y. Let n be a positive integer, C a subalgebra of A n , and h : C → A a homomorphism. By Theorem 5.3, there is a term p : A n → A ℓ in t, and a homomorphism k : Note that k ∈ F, as it is a partial homomorphism of arity ℓ, so k • p belongs to F. Moreover, by Corollary 3.3, π n 1 ↾ C belongs to F, therefore, by Lemma 2.5(3), h = k • p ↾ C belongs to F. Therefore F is the set of all partial operations on A, compatible with A. Moreover, F is, by construction, nitely generated by partial operations of arity at most max{3, ℓ, K}. Clearly, K and 3 are smaller than the bound from the statement of the corollary, as we assumed that A is non-trivial.
It remains to bound the quantity ℓ. Consider the ring R associated to V.
The results follows.
We remark in passing that our corollary provides an additional proof that every nite a ne algebra A is dualizable (even though it is unnecessarily complicated compared to the arguments in [5]). Dualizability of A follows as by the duality compactness theorem [10,11], it su ces to show that the enriched partial hom-clone of A is nitely generated. Our arguments are however not independent, as we rely on Lemmas 2.2, 2.3, and 3.1 from [5].
We are now ready to prove our nal result about the strong dualizability of a ne algebras.
Lemma 5.5. All nite a ne algebras have enough total algebraic operations.
Proof. Let A be an a ne algebra, let ℓ be as in Theorem 5.3 for A = E. We consider ϕ : ω → ω the constant map equal to ℓ. Let n be a positive integer, and let B ≤ C be subalgebras of A n . Denote by ι : By Theorem 5.3 there exist a homomorphism p : A n → A ℓ and a homomorphism k : For each 1 ≤ i ≤ ℓ, denote by π i : A ℓ → A, the canonical projection. Set X = {π i • p | 1 ≤ i ≤ ℓ}, hence X ⊆ Hom(A n , A), moreover |X| ≤ ℓ = ϕ(|B|). Note that Denote by η : C/ ker(p ↾ C) → p(C) the isomorphism induced by p, and by α : B → C/ ker(p ↾ C) the natural homomorphism. We obtain Therefore, A has enough total algebraic operations (cf. De nition 2.6) Theorem 5.6. All nite a ne algebras are strongly dualizable.
Proof. Let A be a nite a ne algebra. By Lemma 5.5, A has enough total algebraic operations, moreover, A is dualizable ( [5], see also the remark a er Corollary 5.4). By Theorem 2.7, A is strongly dualizable.
Our main Theorem 1.1 now follows from the well-known fact that any strongly dualizable algebra is fully dualizable (see for example [2,Theorem 3.2.4]). In our nal result, we provide an explicit bound on the partial functions in the strongly dualizing alter ego.
Theorem 5.7. Let A be a nite a ne algebra with |A| = p α 1 1 · · · p α k k . Then A is strongly dualized by A; P, τ , where τ is the discrete topology on A and P is the set of all compatible partial operations on A of arity at most Proof. As A is strongly dualizable, it is in particular strongly dualized by the strong brute force alter ego A; P, τ of Theorem 2.9. Moreover, by Lemma 2.10 we may replace P with a generating set, and by Corollary 5.4, P is nitely generated by partial operations whose arity is limited by the stated bound. The result follows.

Example
We apply our results to an algebra whose examination was crucial in developing the proofs of the previous sections. Let F 2 be the 2-element eld. Consider the 8-element ring R = F 2 [x, y]/I, where I is the ideal generated by {x 2 , y 2 , xy, yx}. Let A be the module that is obtained by considering R as a module over itself.
By [5], A is dualizable by an alter ego that includes all compatible relations of size 28. By our main result, A is strongly dualizable. A direct application of Theorem 5.7 will result in a very large bound of 702 · 2 81 + 46 on the arities of the partial operations in the alter ego.
By adapting the results of the previous sections to this speci c example, we can show that a lower bound su ces. As A generates a variety V of R-modules, we know the cardinality of the corresponding ring directly. It also means that we can use Lemma 4.4 instead of Lemma 4.5 in our further arguments, which reduces the "Q + 1"-factor in Theorem 5.7 by 1. Also, the estimate Q can be replaced by exact calculations. Instead of the rst factor, we can use the number of nontrivial proper submodules of A, which is easily seen to be 4. The second factor of Q can be replaced by 1 more than the exponent of 2 in the maximal number of homomorphisms from a proper A-submodule G to A. This number can be shown to be 5, meaning that Q + 1 can be replaced with 20. It then follows that we may obtain a strong duality by using an alter ego with "only" the compatible partial operations of arity 559.

Problems
We close with several problems motivated by our results. Problem 1. Which Abelian algebras that do not generate congruence-modular varieties are dualizable? Which are fully and strongly dualizable? Problem 2. Are nilpotent dualizable algebras (from congruence-modular varieties) always fully dualizable? Are they strongly dualizable?
We remark that in many well-behaved classes of algebras, dualizability, full dualizability and strong dualizability coincide. Among nilpotent algebras, the results of [1] show that in the subclass of supernilpotent algebras, all non-Abelian algebras are non-dualizable (and by [1] supernilpotence may be replaced by a slightly weaker condition).

Problem 3.
Can the arity bound in our main theorem be improved upon?
We conjecture that a bound of the form (log 2 |A|) n su ces, for some xed integer n. Problem 4. Which a ne algebras are strongly dualized by some alter ego that is a total structure?

Appendix: Counting homomorphisms and algebras
Lemma 8.1. Let E, F be Abelian groups. Assume that |E| = p α 1 1 · · · p α k k and |F| = p β 1 1 · · · p β k k , where the p i are distinct primes. Then the following statements hold.
(1) The number of subgroups of E is at most p α 2 1 1 · · · p α 2 k k .
(2) |Hom(F, E)| divides p Proof. Note that Abelian subgroups of E are determined by their p i -Sylow subgroups. Denote by E i the p i -Sylow subgroup of E, such that |E i | = p α i i . Each subgroup of E i has a family of generators with α i elements. Therefore E i has at most p  (2) |Hom(F, E)| divides p (α 1 +1)β 1 1 · · · p (α k +1)β k k . Proof. For each c ∈ E, x + c y = t(x, c, y) induces an Abelian group structure on E. There are |E| such structures. Let A be a subalgebra of E, then for c ∈ A, A; + c is a subgroup of E; + c . With Lemma 8.1, the number of subalgebras of E is at most |E| × p  . Let E be an a ne algebra such that |E| = p α 1 1 · · · p α k k , where the p i are distinct primes, and let M = 1 + max 1≤i≤k (α i ). Then E has a generating set with M elements.
The following theorem is a particular case of Kearnes' result in [7], also see [5,Corollary 4.4].

Proof.
A ne algebras are well known to have congruence classes of equal cardinality (see for example [4,Corollary 7.5]). As A is a congruence class of A , the result follows. Lemma 8.6. Let A be an a ne algebra, with |A| = p α 1 1 · · · p α k k where the p i are distinct primes. Set V = Var A. Let L ∈ N. Then |F V (L)| divides p Lα 2 1 +α 1 1 · · · p Lα 2 k +α k k . Proof. We may identify F V (L) with the term clone Clo L (A). Each s : A L → A in Clo L (A) is compatible with t, moreover, Clo L (A) is stable under t, therefore Clo L (A); t is a subalgebra of Hom( A, t L , A, t ); t . Hence |F V (L)| = |Clo L (A)| divides |Hom((A, t) L , (A, t))| which itself divides p Lα 2 1 +α 1 1 · · · p Lα 2 k +α k k by Lemma 8.2 (2).