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equational class (Definition)

Let $ K$ be a class of algebraic systems of the same type. Consider the following “operations” on $ K$:

  1. $ S(K)$ is the class of subalgebras of algebras in $ K$,
  2. $ P(K)$ is the class of direct products of non-empty collections of algebras in $ K$, and
  3. $ H(K)$ is the class of homomorphic images of algebras in $ K$.

It is clear that $ K$ is a subclass of $ S(K),P(K)$, and $ H(K)$.

An equational class is a class $ K$ of algebraic systems such that $ S(K),P(K)$, and $ H(K)$ are subclasses of $ K$. An equational class is also called a variety.

A subclass $ L$ of a variety $ K$ is called a subvariety of $ K$ if $ L$ is a variety itself.

Examples.

Remarks.

  • If $ A,B$ are any of $ H,S,P$, we define $ AB(K):=A(B(K))$ for any class $ K$ of algebras, and write $ A\subseteq B$ iff $ A(K)\subseteq B(K)$. Then $ SH\subseteq HS$, $ PH\subseteq HP$ and $ PS\subseteq SP$.
  • If $ C$ is any one of $ H,S,P$, then $ C^2:=CC=C$.
  • If $ K$ is any class of algebras, then $ HSP(K)$ is an equational class.
  • For any class of algebras, let $ P_S(K)$ be the family of all subdirect products of all non-empty collections of algebras of $ K$. Then $ HSP(K)=HP_S(K)$.
  • 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”. Indeed, a famous theorem of Birkhoff says:
    a class $ V$ of algebras is equational iff there is a set $ \Sigma$ of identities (or equations) such that $ K$ is the smallest class of algebras such that each algebra $ A\in V$ is satisfied by every identity $ e\in \Sigma$. In other words, $ V$ is the set of all models of $ \Sigma$:
    $\displaystyle V=\operatorname{Mod}(\Sigma)=\lbrace A$    is a structure $\displaystyle \mid (\forall e\in \Sigma)\to(A\models e) \rbrace.$

Bibliography

1
G. Grätzer: Universal Algebra, 2nd Edition, Springer, New York (1978).



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See Also: variety of groups

Other names:  variety of algebras, primitive class
Also defines:  variety, subvariety

Attachments:
identity in a class (Definition) by CWoo
implicational class (Definition) by CWoo
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Cross-references: algebra, equations, identities, satisfy, subdirect products, iff, map, canonical, torsion free, complemented, complete, distributive, modular, Noetherian, products, infinite, noetherian rings, Boolean rings, commutative rings, rings, divisible groups, finite groups, cyclic groups, simple groups, abelian groups, subclass, clear, homomorphic images, collections, direct products, algebras, subalgebras, type, algebraic systems, class
There are 36 references to this entry.

This is version 14 of equational class, born on 2007-03-05, modified 2007-12-15.
Object id is 9034, canonical name is EquationalClass.
Accessed 2154 times total.

Classification:
AMS MSC: 08B99 (General algebraic systems :: Varieties :: Miscellaneous)
 03C05 (Mathematical logic and foundations :: Model theory :: Equational classes, universal algebra)
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