is the class of homomorphic images of algebras in .
It is clear that is a subclass of , and .
A subclass of a variety is called a subvariety of if is a variety itself.
If are any of , we define for any class of algebras, and write iff . Then , and .
If is any one of , then .
If is any class of algebras, then is an equational class.
For any class of algebras, let be the family of all subdirect products of all non-empty collections of algebras of . Then .
The reason for call such classes “equational” is due to the fact that algebras within the same class all satisfy a set of “equations”, or “identities (http://planetmath.org/IdentityInAClass)”. Indeed, a famous theorem of Birkhoff says:
a class of algebras is equational iff there is a set of identities (or equations) such that is the smallest class of algebras where each algebra is satisfied by every identity . In other words, is the set of all models of :
- 1 G. Grätzer: Universal Algebra, 2nd Edition, Springer, New York (1978).
|Date of creation||2013-03-22 16:48:02|
|Last modified on||2013-03-22 16:48:02|
|Last modified by||CWoo (3771)|
|Synonym||variety of algebras|