equational class

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

  1. 1.

    S(K) is the class of subalgebrasMathworldPlanetmathPlanetmathPlanetmath of algebrasMathworldPlanetmath in K,

  2. 2.

    P(K) is the class of direct productsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of non-empty collectionsMathworldPlanetmath of algebras in K, and

  3. 3.

    H(K) is the class of homomorphic imagesPlanetmathPlanetmathPlanetmath 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 varietyMathworldPlanetmath.

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



  • If A,B are any of H,S,P, we define AB(K):=A(B(K)) for any class K of algebras, and write AB iff A(K)B(K). Then SHHS, PHHP and PSSP.

  • If C is any one of H,S,P, then C2:=CC=C.

  • If K is any class of algebras, then HSP(K) is an equational class.

  • For any class of algebras, let PS(K) be the family of all subdirect productsPlanetmathPlanetmath of all non-empty collections of algebras of K. Then HSP(K)=HPS(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 “identitiesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/IdentityInAClass)”. Indeed, a famous theorem of Birkhoff says:

    a class V of algebras is equational iff there is a set Σ of identities (or equations) such that K is the smallest class of algebras where each algebra AV is satisfied by every identity eΣ. In other words, V is the set of all models of Σ:

    V=Mod(Σ)={A is a structure (eΣ)(Ae)}.


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