Universal Algebra - Varieties

Varieties

An algebraic structure which can be defined by identities is called a variety, and these are sufficiently important that some authors consider varieties the only object of study in universal algebra, while others consider them an object.

Restricting one's study to varieties rules out:

• Predicate logic, notably quantification, including existential quantification ( ) and universal quantification
• Relations, including inequalities, both and order relations

In this narrower definition, universal algebra can be seen as a special branch of model theory, in which we are typically dealing with structures having operations only (i.e. the type can have symbols for functions but not for relations other than equality), and in which the language used to talk about these structures uses equations only.

Not all algebraic structures in a wider sense fall into this scope. For example ordered groups are not studied in mainstream universal algebra because they involve an ordering relation.

A more fundamental restriction is that universal algebra cannot study the class of fields, because there is no type in which all field laws can be written as equations (inverses of elements are defined for all non-zero elements in a field, so inversion cannot simply be added to the type).

One advantage of this restriction is that the structures studied in universal algebra can be defined in any category which has finite products. For example, a topological group is just a group in the category of topological spaces.