In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a quasivariety is a class of algebraic structure

Algebraic structure

In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...

s generalizing the notion of variety

Variety (universal algebra)

In mathematics, specifically universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities. Equivalently, a variety is a class of algebraic structures of the same signature which is closed under the taking of homomorphic...

 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

Signature (logic)

In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes.Signatures play the same...

 satisfying any of the following equivalent conditions.

1. K is a pseudoelementary class

Pseudoelementary class

In logic, a pseudoelementary class is a class of structures derived from an elementary class by omitting some of its sorts and relations. It is the mathematical logic counterpart of the notion in category theory of a forgetful functor, and in physics of hidden variable theories purporting to...

 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 , where are term

Term (logic)

In mathematical logic, universal algebra, and rewriting systems, terms are expressions which can be obtained from constant symbols, variables and function symbols...

s 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 product

Reduced product

In model theory, a branch of mathematical logic, and in algebra, the reduced product is a construction that generalizes both direct product and ultraproduct....

s.

4. K contains a trivial algebra and is closed under isomorphisms, subalgebras, direct products, and ultraproduct

Ultraproduct

The ultraproduct is a mathematical construction that appears mainly in abstract algebra and in model theory, a branch of mathematical logic. An ultraproduct is a quotient of the direct product of a family of structures. All factors need to have the same signature...

s.

Examples

Every variety

Variety (universal algebra)

In mathematics, specifically universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities. Equivalently, a variety is a class of algebraic structures of the same signature which is closed under the taking of homomorphic...

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