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 quasiidentities, that is, implications of the form formula_1, where formula_2 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 quasiidentity 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.