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

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

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

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

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

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

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

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

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.