##  [HSP Theorem](/hsp-theorem-0) 

 Definition

A characterization (Birkhoff's theorem) of an equational class (a variety) of algebras: a class of algebras of a fixed signature is a variety exactly when it is closed under homomorphic images, subalgebras, and direct products (the closure operators H, S, P).

 

 

 

 

 

 





## Principle

Principle

A class is a variety iff it is H-, S-, and P-closed; equivalently it is the class of all models satisfying a set of identities (equational laws) in the given signature.

 

 

 

 

 





## Demonstration

Demonstration

Example: the class of all groups (same signature: multiplication, inverse, identity) is closed under taking homomorphic images, subgroups, and direct products, therefore it is a variety definable by the group identities; likewise the class of all lattices is closed under H, S, P and so forms a variety.

 

 

 

 

## Misapplication

Misapplication

Applying the HSP criterion to classes defined by arbitrary first-order sentences (with quantifiers or negation) and concluding they are varieties; or checking only two of H, S, P (for example S and P) and declaring a variety without verifying closure under homomorphic images.

 

 

 

 

 





## Consequence

Consequence

When a class is a variety one gets uniform algebraic tools: equational deduction, existence of free algebras, closure under quotients and product constructions, and standard structure theorems such as subdirect representation by subdirectly irreducible members.

 

 

 

 

## Reversal

Reversal

Negating the theorem: a class that fails closure under any one of H, S, or P cannot be an equational variety; for instance the class of fields is not a variety because it fails closure under subalgebras and direct products.

 

 

 

 

 





## Boundary

Boundary

Applies only to algebras of a fixed signature with finitary operations and to classes closed under isomorphism; it does not characterize classes definable solely by first-order axioms that are not equivalent to sets of identities, nor does it apply without an underlying algebraic signature.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Variety (equational class characterized by identities) versus elementary class (axiomatizable by first-order sentences): both are model-theoretic classes but HSP characterizes the former, not the latter; another tension is between HSP and weaker closure properties such as the joint embedding property.

 

 

 

 

 





## Synthesis

Synthesis

The HSP Theorem unifies algebraic and categorical viewpoints: a variety is precisely an equationally axiomatized class and equivalently a class closed under homomorphic images, subalgebras and direct products, which yields canonical constructions (quotients, substructures, products) and underpins free objects and subdirect decompositions.