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. Definition

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. K is a pseudoelementary class closed under subalgebras and direct products. 2. K is the class of all models of a set of quasi-identities, that is, implications of the form , where  are terms built up from variables using the operation symbols of the specified signature. 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. Examples

Every variety is a quasivariety by virtue of an equation being a quasi-identity for which n = 0. The cancellative semigroups form a quasivariety. 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. References

Universal algebra