Birkhoff's Theorem
Let 
and 
be two algebras
over the same signature 
, with carriers
and 
, respectively (cf.
universal algebra). 
is a subalgebra
of 
if 
and
every function of 
is the restriction of the respective
function of 
on 
.
The (direct) product of algebras 
and 
is an algebra
whose carrier is the Cartesian product of 
and 
and such that for
every 
and all 
and all 
,
 |
A nonempty class 
of algebras over the same signature
is called a variety if it is closed under subalgebras,
homomorphic images, and Cartesian products over arbitrary families of structures
belonging to the class.
A class of algebras is said to satisfy the identity 
if this identity
holds in every algebra from this class. Let 
be a set of identities
over signature 
. A class 
of algebras over
is called an equational class if
it is the class of algebras satisfying all identities from 
. In this case,
is said to be axiomatized by 
.
Birkhoff's theorem states that 
is an equational
class iff it is a variety.
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