The class of hyperalgebras forms a variety of algebras of
type (0, 0, ..., 2, 3, 4, ...) in the sense of universal algebra,
therefore free hyperalgebra T(F) over any nonempty set F (of functional
variables) exists. The construction of T(F) given below is very simple,
which leads to a canonical definition of hyperidentities.
Let T(F) be the smallest set containing X = {x_1, x_2, ...} and F such that
if f is in M and M_{1}, ,,,, M_{m} are in T(F) then f[M_{1}, ,,,, M_{m}]
= f[M_{1}, ..., M_{m}, M_{m}, ...]
is in T(F), where f[M_{1}, ,,,, M_{m}] is viewed as an
abbreviation of f[M_{1}, ..., M_{m}, M_{m}, ...].
Thus we have f[M_{1}, ,,,, M_{m}] = f[M_{1}, ..., M_{m},
M_{m}].
Define M[N_{1}, ..., N_{n}] in T(F) inductively on M:
1. x_{i}[N_{1}, ..., N_{n}] = N_{i} if i < n
+ 1 and x_{i}[N_{1}, ..., N_{n}] = N_{n}
otherwise. 2. f[N_{1}, ..., N_{n}] = f[N_{1}, ..., N_{n }
, N_{n} ...] for any f in F.
3. (f[M_{1}, ,,,, M_{m}])[N_{1}, ..., N_{n}]
= f[M_{1}[N_{1}, ..., N_{n}], ..., M_{m}[N_{1},
..., N_{n}]].
Then it is easy to see that T(F) is a free hyperalgebra over F.
Definition. A hyperidentity is an equation M = N with terms M, N
in T(F). We say M = N is a finitary hyperidentity if M, N are finitary
elements of T(F).
We say a hyperalgebra A satisfies a hyperidentity M = N (or M = N is a
hyperidentity of A) if P(M) = P(V) for any homomorphism P: T(F) > A. We say
a variety V satisfies a hyperidentity M = N (or M = N is a hyperidentity of
V) if the free algebra T(V) in V over X satisfies the hyperidentity M = N.
To simplify notations we often let x = x_{1}, y = x_{2},
z = x_{3}, ... The functional variables in F will be denoted
by f, g, h, ...
Example. The variety of lattices satisfies the following finitary
hyperidentities:
a. f[x, x] = x.
b. f[f[x, y], y] = f[x, y].
c. f[f[x, y], z] = f[x, f[y, z]].
If E is a set of hyperidentities denote by Hyp(E) the class of hyperalgebras
satisfying each hyperidentity of E. A class K of hyperalgebras is called
equationally definable if there is a set E of hyperidentities such that K =
Ha(E).
If K is a class of hyperalgebras denote by Hid_{F}(K) the class of
hyperidentities in T(F) satisfied in every hyperalgebra of K. Then Hid_{F}(K)
is a fully invariant congruence relation in the free hyperalgerbra T(F).
Clearly if K consists of finitary hyperalgebras then Hid_{F}(K) is
generated by finitary hyperidentites.
Definition. 1. A class K of hyperalgebras is called a hypervariety if
it is closed under the formation of subalgebras, homomorphic images, and
direct products.
2. Let F = {f_{1}, f_{2}, ...}. A subset E is called a
hypertheory if E = Hid_{F}(K) for some class of hyperalgebras.
The class of hypervarieties forms a complete lattice under intersection. We
say a variety V belongs
to (resp. generates) a hypervariety if the hyperalgebra T(V) belongs to (resp.
generates) the
hypervariety. We say that a hypervariety is locally finitary if it is
generated by finitary hyperalgebras.
Theorem (Birkhoff's Theorem) 1. A class of hyperalgebras is
equationally definable iff it is a hypervariety. 2. The complete lattice of hypervarieties is dually isomorphic to the
complete lattice of hypertheories.
Remark. Assume (F, α) is any algebraic
similarity type, where F is a nonempty set and α
is an arity function which assigns to each f in F a positive integer α(f).
Then the term algebra W(F) of type (F, α) over X is naturally a finitary
hyperalgebra because it is the free algebra of type (F, α) over X. The
hyperlagebra W(F) may be viewed as a finitary subalgebra of T(F) generated
by the subsets { f[x_{1}, ..., x_{n}]  f is in F with arity n}. Thus an
identity M = N with terms in W(F) may be viewed as a finitary
hyperidentity in T(F). Since W(F) is a retraction of T(F), any homomorphism
from W(F) to a hyperalgebra A extends to a homomrophism from T(F) to A. It
follows that A satisfies a hypervariety M = N in T(F), where M, N are in
W(F), iff P(M) = P(N) for any homomorphism P: W(F) > A. Now any
homomorphism P: W(F) > A is uniquely defined by a function p: F > A such
that p(f) has a finite rank n for any f in F with arity n. Since any
finitary hyperidentity in T(F) may be viewed as a hyperidentity in W(F) for
a suitable algebraic similarity type (F, α), we see that our approach to the
theory of hyperidentity is equivalent to the tradition approach given in
literature.
