##  [Quasivariety](/quasivariety-0) 

 Definition

A class of algebras axiomatizable by a set of universal Horn sentences; equivalently, a class closed under taking subalgebras, direct products and ultraproducts (and containing the trivial algebra when required).

 

 

 

 

 

 





## Principle

Principle

Characterize algebraic classes by conditional universal identities rather than pure identities; closure under substructures and products plus preservation by ultraproducts is the organizing idea.

 

 

 

 

 





## Demonstration

Demonstration

The class of torsion-free abelian groups is axiomatizable by the family of universal Horn sentences (n·x = 0) → x = 0 for each natural n>0, so it is a quasivariety though not a variety.

 

 

 

 

## Misapplication

Misapplication

Assuming a quasivariety is closed under all homomorphic images as a variety is; homomorphic images need not remain in a quasivariety unless further axioms force it.

 

 

 

 

 





## Consequence

Consequence

Quasivarieties admit relative free algebras (free objects in the class) and are stable under model-theoretic constructions such as ultraproducts; they precisely capture the model classes of universal Horn theories.

 

 

 

 

## Reversal

Reversal

A variety is the stronger notion obtained when the axioms can be taken as pure equations; varieties are quasivarieties that are also closed under homomorphic images.

 

 

 

 

 





## Boundary

Boundary

Includes only classes definable by universal Horn sentences; excludes properties requiring existential axioms or full first-order conditions that break subalgebra or product closure.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Between quasivariety and pseudovariety: quasivarieties allow infinite algebras and ultraproduct closure, while pseudovarieties restrict attention to finite algebras and closure under finite products and quotients, leading to different applications.

 

 

 

 

 





## Synthesis

Synthesis

A quasivariety is the model-theoretic closure class generated by universal Horn axioms: it generalizes varieties by admitting conditional constraints, yields relative free objects and is the natural semantic home for universal Horn theories.