equational class (original) (raw)

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.

Remarks.

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  If A,B are any of H,S,P, we define A⁢B⁢(K):=A⁢(B⁢(K)) for any class K of algebras, and write A⊆B iff A⁢(K)⊆B⁢(K). Then S⁢H⊆H⁢S, P⁢H⊆H⁢P and P⁢S⊆S⁢P.
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  If C is any one of H,S,P, then C2:=C⁢C=C.
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  If K is any class of algebras, then H⁢S⁢P⁢(K) is an equational class.
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  For any class of algebras, let PS⁢(K) be the family of all subdirect products of all non-empty collections of algebras of K. Then H⁢S⁢P⁢(K)=H⁢PS⁢(K).
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References