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]

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 $s_1 \approx t_1 \land \ldots \land s_n \approx t_n \rightarrow s \approx t$, where $s, s_1, \ldots, s_n,t, t_1, \ldots, t_n$ 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.

Every class of ordered algebras is a quasivariety, since the partial order axioms are quasiidentities.^[2]

References

1. Stanley Burris; H.P. Sankappanavar (1981). A Course in Universal Algebra. Springer-Verlag. ISBN 0-387-90578-2.
2. Viktor A. Gorbunov (1998). Algebraic Theory of Quasivarieties. Siberian School of Algebra and Logic. Plenum Publishing. ISBN 0-306-11063-6.