Varieties

A variety, $V$ is a class of Algebras which is defined to be the class of ALL algebras which satisfy a set of predefined laws.

Examples include rings, groups, magmas but NOT fields etc. The reason fields dont count is because they specify “non 0” as a component of a definition which is not allowed.

Variety Generation

A variety can be generated from a specific algebra by generalizing all the laws that are satisfied by an algebra. This can be a difficult task.

Much like how groups can be generated by elements varieties can be generated by algebras.

THEOREM: The variety ${\mathbb{V}}_A$ generated by an algebra $A$ is the smallest class of algebras such that

1. it contains the algebra in question
2. it is closed under sub algebras
3. it is closed under product algebras
4. it is closed under taking homomorphic images (laws are carried across a homomorphism) Closure in this case means one can apply the same laws under the different constructions. Meaning it’s in the variety as well.

Example

The variety generated by the group $(\mathbb{Z},+,-,0)$ (It also has the unary operation $-$ which inverts elements and the nullary operation $0$ which is the identity.)

Luna claims ${\mathbb{V}}_{(\mathbb{Z},+,-,0)}$ is the variety of abelian groups. This is equivalent to claiming this group has no laws which are not shared with all abelian groups. So this group is the primordial origin abelian group effectively from which all laws flow out and are appended.