Variety (universal algebra)

In universal algebra, a variety of algebras or equational class refers to a group of all algebraic structures that share the same type of operations and follow specific rules, called identities. These rules are equations that must always be true for every structure in the variety. For instance, groups, abelian groups, rings, and monoids are all examples of varieties because they follow certain identities that define their behavior.

One important idea in this area is Birkhoff's theorem, which tells us that a class of algebraic structures belongs to a variety if it stays the same when we take homomorphic images, subalgebras, and products. This helps mathematicians understand how different algebraic systems relate to each other.

In more advanced mathematics, especially in category theory, a variety of algebras can also be viewed as a special kind of category. These are known as finitary algebraic categories, linking algebra and category theory in powerful ways. Additionally, there is a related idea called a covariety, which deals with coalgebraic structures instead of algebras. This whole area helps us organize and understand the many different 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 collection of operations, where each operation has a specific number of inputs, called its arity. Using this signature and a set of variables, we can build words, which are like formulas made from these operations and variables.

A theory includes a signature, some variables, and a set of rules called equational laws. These rules state that two words are equal. An algebra based on this theory is a set along with specific ways to apply each operation, following all the rules in the theory. When we look at all such algebras together, we call this collection a variety of algebras.

This theory also helps us understand how different algebras relate to each other through homomorphisms, which are special mappings that preserve the operations’ behavior.

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 class of algebraic structures that share the same type of operations and follow certain rules, called identities. Garrett Birkhoff showed that a class of these structures is a variety if it stays the same when we take homomorphic images, subalgebras, and products. This important result is known as Birkhoff's variety theorem or the HSP theorem, where H, S, and P stand for homomorphism, subalgebra, and product.

One part of this theorem is easy to understand: if a class of algebras follows some identities, it will stay the same under these operations. Proving the opposite—that if a class stays the same under these operations, it must follow some identities—is more challenging. For example, fields do not form a variety because the product of two fields is not always a field.

Subvarieties

A subvariety is a smaller group within a bigger group of algebraic structures that follows the same rules and also forms its own group of structures. For example, the class of abelian groups is a subvariety of the variety of groups because it follows the same rules without changing the way the structures are defined.

The finitely generated abelian groups do not form a subvariety because they do not satisfy the conditions to form a variety on their own. When we think of varieties and their homomorphisms as categories, a subvariety is a full subcategory, meaning 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, such as certain types of Boolean algebras and sigma algebras.

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.