Quasivariety

Quasivariety

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

In mathematics, 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 "s"1 = "t"1 … "s""n" = "t""n" → "s" = "t" where "s" and "t" are terms built up from variables using the operation symbols of the specified signature.

3. "K" is closed under isomorphisms, subalgebras, and reduced products and contains the trivial algebra.

4. "K" is closed under isomorphisms, subalgebras, direct products, and ultraproducts and contains the trivial algebra.

Examples

Every variety is a quasivariety by virtue of an equation being a quasiidentity for which "n" = 0.

References

Stanley Burris and H.P. Sankappanavar, "A Course in Universal Algebra", Springer-Verlag, 1981. ISBN 0-387-90578-2, ISBN 3-540-90578-2.


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