A variety in universal algebra is a class of algebras of the same signature closed under homomorphic images (H), subalgebras (S), and arbitrary direct products (P); equivalently, a class axiomatizable by a set of identities (equations).

Principle

Equational closure: identities determine the class, and closure under H, S, and P guarantees that syntactic equations translate to robust algebraic closure properties (Birkhoff's HSP theorem).

Demonstration

Groups (same signature: multiplication, inverse, unit) form a variety because they are defined by identities and are closed under homomorphic images, subgroups, and products; fields do not form a variety because closure under products or homomorphic images fails for field axioms.

Misapplication

Calling any class closed under subalgebras and products a variety without checking closure under homomorphic images, or confusing varieties with quasivarieties (which are closed under S, P, and ultraproducts but axiomatizable by quasi-identities).

Consequence

Varieties admit free algebras, equational reasoning and term rewriting, and universal constructions; they support a strong structure theory and categorical equivalences with algebraic monads in Set.

Reversal

The inverse concept is a class defined by non-equational constraints (for example, requiring existence of inverses by an existential axiom) which may be closed under fewer operations and fails HSP closure.

Boundary

Requires a single fixed signature and finitary operations; does not capture relational structures or classes defined by infinitary or existential conditions. Multi-sorted or infinitary generalizations require different closure theorems.

Semantic Tension

Competes with the notion of quasivariety and elementary class: varieties are purely equational while elementary classes may require first-order axioms; some natural algebraic classes lie strictly between these notions.

Synthesis

A variety is the algebraic manifestation of equational logic: a signature plus identities generate a class stable under homomorphisms, substructures and products, enabling free objects and algebraic manipulation strictly governed by identities.