equational class

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

1. 1.

$S(K)$ is the class of subalgebras of algebras in $K$,

2. 2.

$P(K)$ is the class of direct products of non-empty collections of algebras in $K$, and

3. 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 (http://planetmath.org/IdentityInAClass)”. 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 where each algebra $A\in V$ is satisfied by every identity $e\in\Sigma$. In other words, $V$ is the set of all models of $\Sigma$:

 $V=\operatorname{Mod}(\Sigma)=\{A\mbox{ is a structure }\mid(\forall e\in\Sigma% )\to(A\models e)\}.$

References

• 1 G. Grätzer: , 2nd Edition, Springer, New York (1978).
 Title equational class Canonical name EquationalClass Date of creation 2013-03-22 16:48:02 Last modified on 2013-03-22 16:48:02 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 19 Author CWoo (3771) Entry type Definition Classification msc 08B99 Classification msc 03C05 Synonym variety of algebras Synonym primitive class Related topic VarietyOfGroups Defines variety Defines subvariety