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.
