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.