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.

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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. ''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.

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.

References

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Universal algebra