# Quasivariety

In mathematics, a **quasivariety** is a class of algebraic structures generalizing the notion of variety by allowing equational conditions on the axioms defining the class.

A *trivial algebra* contains just one element. A **quasivariety** is a class *K* of algebras with a specified signature satisfying any of the following equivalent conditions.^{[1]}

1. *K* is a pseudoelementary class closed under subalgebras and direct products.

3. *K* contains a trivial algebra and is closed under isomorphisms, subalgebras, and reduced products.

4. *K* contains a trivial algebra and is closed under isomorphisms, subalgebras, direct products, and ultraproducts.

Every variety is a quasivariety by virtue of an equation being a quasiidentity for which *n* = 0.

Let *K* be a quasivariety. Then the class of orderable algebras from *K* forms a quasivariety, since the preservation-of-order axioms are Horn clauses.^{[2]}