Quasivariety Explained

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.

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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 quasi-identities, that is, implications of the form

s _{1} ≈ t _{1} \land \ldots \land s _{n} ≈ t _{n} → s ≈ 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.

1. K contains a trivial algebra and is closed under isomorphisms, subalgebras, and reduced products.
2. 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 .

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.^[2]

Notes and References

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