##  [Variety (Universal Algebra)](/variety-universal-algebra-1) 

 Definition

A class of algebras of a fixed signature that is closed under homomorphic images, subalgebras, and arbitrary direct products; equivalently, the class of all models satisfying a given set of algebraic identities (equations).

 

 

 

 

 

 





## Principle

Principle

Equational axiomatizability: a set of identities determines a class closed under the H (homomorphic images), S (subalgebras), and P (direct products) operators; conversely, any class with these closure properties is definable by identities over the signature.

 

 

 

 

 





## Demonstration

Demonstration

The class of all groups presented for a group signature with the usual group identities (associativity, identity element, inverse laws expressed as equations with a unary inverse operation) is a variety: it is closed under taking subgroups, quotient groups (homomorphic images), and direct products of groups.

 

 

 

 

## Misapplication

Misapplication

Assuming a variety can be characterized by arbitrary first‑order sentences or by closure under ultraproducts alone; this confuses equational definability with broader model‑theoretic properties and can lead to wrong closure expectations.

 

 

 

 

 





## Consequence

Consequence

Varieties admit free objects on any generating set, have congruence‑based structure theory, and allow equational deduction: checking an identity in the variety reduces to term rewriting and syntactic manipulations within generated free algebras.

 

 

 

 

## Reversal

Reversal

A class closed only under subalgebras and products but not homomorphic images (or axiomatizable by implications rather than identities) is not a variety but a weaker class such as a quasivariety; reversing the closure axioms yields strictly different model‑theoretic behavior.

 

 

 

 

 





## Boundary

Boundary

Applies to classes definable by purely equational axioms in a fixed finitary signature; it excludes classes requiring implication between atomic formulas, existential quantifiers, or other non‑equational constraints, and excludes properties that fail closure under arbitrary direct products.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension exists between varieties (equational closures) and quasivarieties or elementary classes: the same intuitive family of structures may be equationally describable or instead require Horn or full first‑order axiomatizations, affecting closure operations and free constructions.

 

 

 

 

 





## Synthesis

Synthesis

A variety is precisely an equationally defined class of algebras for a signature, characterized by closure under homomorphic images, subalgebras, and arbitrary products; this equivalence ties syntactic identities to robust algebraic closure properties and to the existence of free algebras.