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]