**Definition.** A hyperalgebra is a nonempty set A
together with a sequence X = {x_{1}, x_{2,} ...} of elements
of A and a sequence S = {s_{1}, s_{2}, ...} of operations s_{n}:
A^{n+1} -> A, which satisfies the following three axioms for M, M_{1},..., M_{m},
N_{1}, ..., N_{n} in A.
Write M[M_{1}, ..., M_{n}] for s_{n}(M, M_{1},
..., M_{n}).
A1. x_{n}[M_{1}, ..., M_{m}] = M_{n} if n
≤ m.
A2. (M[M_{1}, ..., M_{m}])[N_{1}, ..., N_{n}]
= M[M_{1}[N_{1}, ..., N_{n}], ..., M_{m}[N_{1},
..., N_{n}]].
A3. M[M_{1}, ..., M_{m}] = M[M_{1}, ..., 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[x_{1},
..., x_{n}], We say M has a rank 0 (or M is closed) if M[x_{1}]
= M[x_{2}] = 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 = {x_{1}, x_{2}, ...}. Then T(V) is naturally a
finitary hyperalgebra such that M[M_{1}, ..., M_{m}] is the image of
M under the endomorphism on T(V) sending x_{i} to M^{i} 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
= {u_{1}, u_{2},...} of operations u_{n}: A x D^{n}
-> D, which satisfying the following axioms for M, M_{1}, ..., M_{m}
in A and a_{1}, ..., a_{n} in D:
Write M[a_{1}, ..., a_{n}] for u_{n}(M, a_{1},
..., a_{n}).
B1. x_{n}[a_{1}, ..., a_{m}] = a_{n} if n ≤
m.
B2. (M[M_{1}, ..., M_{m}])[a_{1}, ..., a_{n}]
= M[M_{1}[a_{1}, ..., a_{n}], ..., M_{m}[a_{1},
..., a_{n}]].
B3. M[a_{1}, ..., a_{n}] = M[a_{1}, ..., 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 [a_{1}, ..., 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[M_{1}, ..., M_{m}]
being defined by M[M_{1}, ..., M_{m}][a_{1}, a_{2},
...] = M[M_{1}[a_{1}, a_{2}, ...], ..., M_{n}[a_{1},
a_{2}, ...], M_{n}[a_{1}, a_{2}, ...], ...].
**
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)[M_{1}, ...,
M_{n},_{
}M_{n}, M_{n},_{
}...] = M[M_{1}, ..., M_{n}]. If A is finitary then h(A)
is finitary and h is injective as M[x_{1}, ..., 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. |