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]

K is a pseudoelementary class closed under subalgebras and direct products.
K is the class of all models of a set of quasi-identities, 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.
K contains a trivial algebra and is closed under isomorphisms, subalgebras, and reduced products.
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 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.^[2]

References

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

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