A class of algebras axiomatizable by universal Horn sentences (quasi‑identities); equivalently a class closed under subalgebras, direct products, and ultraproducts (and isomorphisms), but not necessarily under homomorphic images.

Principle

Weaken the equational requirement by permitting universal Horn implications between atomic formulas: axioms have the form 'conjunction of equations implies an equation'; this yields closure under S (subalgebras), P (products), and ultraproducts while allowing restrictions that are not purely equational.

Demonstration

The class of torsion‑free abelian groups can be expressed by the family of quasi‑identities 'for each n>0, n·x = 0 implies x = 0' together with abelian group identities; that class is closed under taking subgroups and direct products and under ultraproducts, so it exemplifies a quasivariety that is not a variety because quotients need not remain torsion‑free.

Misapplication

Treating every quasivariety as a variety and expecting closure under homomorphic images will yield counterexamples: a quotient of a structure satisfying the Horn axioms may violate the implicational conditions and fall outside the class.

Consequence

Quasivarieties retain many algebraic tools (product and substructure constructions, some free‑object notions in restricted senses) and provide the natural setting for properties expressible by implication of equations, bridging equational algebra and more general universal theories.

Reversal

Requiring full equational axioms (identities only) strengthens a quasivariety to a variety and restores closure under homomorphic images and the full machinery of equational free constructions; reversing in the other direction (allowing arbitrary first‑order sentences) loses the uniform Horn‑closure properties.

Boundary

Includes classes definable by universal Horn theory over a fixed signature; it excludes classes that require existential quantifiers, disjunctions at top level, or non‑Horn universal sentences, and it does not guarantee closure under arbitrary homomorphic images.

Semantic Tension

The main tension is between the expressive convenience of Horn implications (allowing conditional identities) and the algebraic strength of pure identities: some natural families are quasivarieties but not varieties, producing ambiguity about which closure principles apply.

Synthesis

A quasivariety is the collection of models of universal Horn axioms: it generalizes varieties by admitting conditional equations, yielding closure under subalgebras, products, and ultraproducts but typically lacking closure under arbitrary homomorphic images, and thus forms the intermediary between equational and full first‑order classes.