Zhaohua Luo

Abstract

Definition. A hyperalgebra is a nonempty set A together with a sequence X = {x₁, x_2, ...} of elements of A and a sequence S = {s₁, s₂, ...} of operations s_n: Aⁿ⁺¹ -> A, which satisfies the following three axioms for M, M₁,..., M_m, N₁, ..., N_n in A.
Write M[M₁, ..., M_n] for s_n(M, M₁, ..., M_n).
A1. x_n[M₁, ..., M_m] = M_n if n ≤ m.
A2. (M[M₁, ..., M_m])[N₁, ..., N_n] = M[M₁[N₁, ..., N_n], ..., M_m[N₁, ..., N_n]].
A3. M[M₁, ..., M_m] = M[M₁, ..., M_m, M_m].

The class Hyp of hyperalgebras forms a variety. Therefore the category Hyp is complete and cocomplete, with free algebras over arbitrary set.

We say an element M of a hyperalgebra A has a finite rank n > 0 if M = M[x₁, ..., x_n], We say M has a rank 0 (or M is closed) if M[x₁] = M[x₂] = M. An element M in A is called  finitary if it has a finite rank. If n is a nonnegative integer denote by F_n(A) the set of elements of rank n  of A.   

Denote by Fin(A) the set of finitary elements of A. It is the largest finitary subalgebra of A. Since any homomorphism of hyperalgebras preserves finitary elements, the category FHyp of finitary  hyperalgebras is a coreflective subcategory of Hyp. Therefore FHyp is also complete and cocomplete. But FHyp is not algebraic over Set, because it has no free algebra over any nonempty set.

Note that a hyperalgebra A is finitary iff it is generated by a set of finitary elements.

We have the following basic facts:
1, Any subalgebra of a finitary hyperalgebra is finitary.
2. Any homomorphic image of a finitary hyperalgebra is finitary.
3. The Direct product of a collection K of finitary hyperalgebras  is finitary iff K is a finite set.
Thus if G is the direct product of finitary hyperalgebras for an infinite set K then Fin(G) is the direct product
of these finitary hyperalgebras in FHyp.

Example. Let V be a variety and let T(V) be the free algebra of V over X = {x₁, x₂, ...}. Then T(V) is naturally a finitary hyperalgebra such that M[M₁, ..., M_m] is the image of M under the endomorphism on T(V) sending x_i to Mⁱ if i ≤ m and to M_m otherwise.

Let A be a hyperalgebra. A model of A is a set D together with a sequence U = {u₁, u₂,...}  of operations u_n: A x Dⁿ -> D, which satisfying the following axioms for M, M₁, ..., M_m in A and a₁, ..., a_n in D:
Write M[a₁, ..., a_n] for u_n(M, a₁, ..., a_n).
B1. x_n[a₁, ..., a_m] = a_n if n ≤ m.
B2. (M[M₁, ..., M_m])[a₁, ..., a_n] = M[M₁[a₁, ..., a_n], ..., M_m[a₁, ..., a_n]].
B3. M[a₁, ..., a_n] = M[a₁, ..., a_n, a_n ].

Define homomorphisms of models of A in an obvious way. Denote by Mod(A) the category of models of A.

Example. Note that A is a model of A. For any nonnegative integer n the set F_n(A)  of elements of rank n of A is also a model of A. In fact F_n(A)  is the free model of A of rank n.

Theorem. 1. If V is a variety and T(V) is the free algebra of V over X, then T(V) is naturally a finitary hyperalgebra algebra, and V is equivalent to Mod(T(V)) as concrete categories over Set .
2. If A is a hyperalgebra then the class Mod(A) forms a variety. If A is a finitary  hyperalgebra then A is isomorphic to T(Mod(A)).
3. The correspondences A -> Mod(A) and V -> T(V) establish an equivalence between the category FHyp of finitary hyperalgebras and the opposite of the category of varieties.

Example. For any nonempty set S let [S*] be the set of all infinite sequences [a₁, ..., a_n, a_n, a_n  ...] (n > 0) of elements of S. Let H(S) be the set of all  functions from [S*] to S. Let x_i : [S*] -> S be the i-th projection. Then H(S) is naturally a hyperalgebra with M[M₁, ..., M_m] being defined by M[M₁, ..., M_m][a₁, a₂, ...] = M[M₁[a₁, a₂, ...], ..., M_n[a₁, a₂, ...], M_n[a₁, a₂, ...], ...].

Definition. H(S) is called a primal hyperalgebra. F(H(S))  is called a finitary primal hyperalgebra.

Any hyperalgebra A determines a homomorphism h: A -> H(A) sending each M in A to the function h(M): [A*] -> A  given by h(M)[M₁, ..., M_n, M_n, M_n,  ...] = M[M₁, ..., M_n]. If A is finitary then h(A) is finitary and h is injective as M[x₁, ..., x_n] = M for some n > 0.. Since h(A) is a subalgebra of F(H(A)),  we have the following

Theorem (Cayley's theorem for hyperalgebras) Any finitary hyperalgebra is isomorphic to a subalgebra of a finitary primal hyperalgebra.

Part I  II  III