Quasivariety

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A quasivariety is a class of algebraic structures generalizing the notion of variety by allowing equational conditions on the axioms defining the class.

Contents 1 Definition 2 Examples 3 References

[edit] 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 s₁ = t₁ … 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.

[edit] Examples

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

[edit] 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.