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单词 EquationalClass
释义

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.

Examples.

  • In the variety of groups, the classes of abelian groupsMathworldPlanetmath is equational. However, the following classes are not: simple groupsMathworldPlanetmathPlanetmath, cyclic groupsMathworldPlanetmath, finite groupsMathworldPlanetmath, and divisible groups.

  • In the variety of rings, the classes of commutative rings and Boolean ringsMathworldPlanetmath are varieties. Most classes of rings, however, are not equational. For example, the class of Noetherian ringsMathworldPlanetmath is not equational, as infinite products of Noetherian rings are not Noetherian.

  • In the variety of lattices, the classes of modular lattices and distributive lattices are equational, while completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath lattices and complementedPlanetmathPlanetmath lattices are not.

  • The class of Heyting algebrasMathworldPlanetmath is equational, and so is the subclass of Boolean algebrasMathworldPlanetmath.

  • The class of torsion free abelian groups is not equational. For example, the homomorphic image of under the canonical map n is not torsion free.

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 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)}.

References

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

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更新时间:2025/5/4 17:34:27