Variety (universal algebra)

Variety (universal algebra)

In universal algebra, a variety of algebras or equational class is a group of all algebraic structures that use the same operations and follow special rules. These rules are equations that are always true for every structure in the variety. For example, groups, abelian groups, rings, and monoids are all varieties because they follow certain rules that define how they work.

One key idea is Birkhoff's theorem. It says a class of algebraic structures is a variety if it stays the same when we look at homomorphic images, subalgebras, and products. This helps mathematicians see how different algebraic systems are connected.

In more advanced math, especially in category theory, a variety of algebras can also be seen as a special type of category. These are called finitary algebraic categories, linking algebra and category theory in useful ways. There is also a related idea called a covariety, which looks at coalgebraic structures instead of algebras. This area helps us organize and understand the many types of mathematical structures that exist.

Terminology

A variety of algebras is a special group of mathematical structures that follow certain rules. It is different from an algebraic variety, which deals with solving sets of equations.

The term "variety of algebras" talks about algebras in a broad way, linked to universal algebra. It can also mean a specific type of algebra related to algebra over a field, like a vector space with a special kind of multiplication called bilinear.

Definition

A signature is a list of operations. Each operation has a set number of inputs, called its arity. Using this signature and some variables, we can make words. These words are like formulas built from the operations and variables.

A theory includes a signature, variables, and rules called equational laws. These rules say when two words are equal. An algebra based on this theory is a group with ways to use each operation, following all the rules. When we put all these algebras together, we call it a variety of algebras.

This theory also helps us see how different algebras are connected through homomorphisms. These are special maps that keep the operations working the same way.

Examples

Some types of mathematical structures are called "varieties." This means they follow certain rules, or identities.

For example, semigroups are structures with one operation that follows the associative law: x (y z) = (x y) z.

Groups are another example. They have three operations: multiplication, identity, and inversion. They follow identities like associativity, identity (1 x = x 1 = x), and inversion (x x⁻¹ = x⁻¹ x = 1).

Rings also form a variety with their own set of operations and identities. However, fields do not form a variety because the rule that every non-zero element has an inverse cannot be written as a simple identity. Similarly, cancellative semigroups do not form a variety but instead form something called a quasivariety.

Birkhoff's variety theorem

In universal algebra, a variety is a group of math structures that use the same operations and follow special rules, called identities. Garrett Birkhoff showed that a group of these structures is a variety if it stays the same when we make certain changes, take smaller parts, and combine structures. This key finding is called Birkhoff's variety theorem or the HSP theorem, where H, S, and P mean homomorphism, subalgebra, and product.

One part of this theorem is simple: if a group of math structures follows some identities, it will stay the same with these changes. Proving the other part—that if a group stays the same with these changes, it must follow some identities—is harder. For example, fields are not a variety because when you combine two fields, the result is not always a field.

Subvarieties

A subvariety is a smaller group inside a bigger group of algebraic structures. It follows the same rules and also makes its own group of structures. For example, the class of abelian groups is a subvariety of the variety of groups. This is because it follows the same rules without changing how the structures are defined.

The finitely generated abelian groups do not form a subvariety. They do not meet the conditions to form a variety on their own. When we see varieties and their homomorphisms as categories, a subvariety is a full subcategory. This means it includes all the homomorphisms between its objects that the larger variety has.

Main article: finitely generated abelian groups

Main article: full subcategory

Free objects

In a variety of algebras, there is a special kind of algebra called a free algebra linked to any set of elements. This free algebra includes all possible combinations of the elements, following the rules of the variety.

This idea is similar to free groups and free modules in other areas of mathematics. It means that any algebra in the variety can be thought of as coming from a free algebra.

Category theory

Category theorists use different ways to describe the same kinds of algebras as varieties, such as finitary monads and Lawvere theories. These methods help us understand how algebras are built from basic sets and how they relate to each other.

When we use monads, we can create a special type of category called a finitary algebraic category. These categories help us study algebras with operations that involve only a finite number of elements. More general algebraic categories can include algebras with operations that involve infinitely many elements.

Pseudovariety of finite algebras

A pseudovariety is a group of special types of mathematical structures called algebras. These structures follow certain rules and can be combined in limited ways. Some people use the term "variety of finite algebras" when all the structures in the group are small and limited in size.

Pseudovarieties are very useful when studying small structures called finite semigroups. They also help in the study of patterns in language and symbols, known as formal language theory. A famous result called Eilenberg's theorem shows a connection between certain language patterns and these special groups of algebras.