Universal algebra

Universal algebra is a part of mathematics. It studies different types of algebraic structures. Instead of focusing on one specific type, like groups or rings, it looks at many types and how they are connected. This helps mathematicians see patterns and properties that many algebraic systems share. By studying structures in a broad way, universal algebra offers useful tools and ideas for many areas of math and logic.

Basic idea

Main article: Algebraic structure

Not to be confused with Algebra over a field.

In universal algebra, an algebra is a set with some actions you can do on the things in that set. These actions can mix the things in different ways.

Arity

Main article: Arity

An n-ary operation is a rule that takes n things and gives back one thing. For example, a 0-ary operation is just a fixed thing, like a constant. A 1-ary operation takes one thing and returns one thing, like a symbol in front of it. A 2-ary operation takes two things and returns one thing, often shown between the two things. Operations that take more than two things are usually written with the things listed in parentheses.

Varieties

Main article: Variety (universal algebra)

Not to be confused with Algebraic variety.

A group of math structures defined by equations is called a variety or equational class. This means we only use equations to describe these structures. We don't use ideas like "for all" or "there exists" beyond simple equations.

The study of these equational classes is a special part of model theory. It looks at structures with operations, like functions, and uses only equations to describe them. For example, ordered groups don't fit here because they use an ordering idea, not just equations. Also, the class of fields isn't an equational class because you can't write all field rules as simple equations.

One good thing about this way of studying is that these structures can be looked at in any category that has finite products. For example, a topological group is just a group inside the group of topological spaces.

Examples

Most common algebra systems in math are examples of varieties. Their usual rules might seem different because they often use ideas like "for all" or "there exists".

For example, a group is usually defined with three rules: associativity, having an identity element, and each element having an inverse. In universal algebra, we change the definition to use only equations. We add extra operations: a special element for the identity, and an operation to find the inverse of an element. With these, all group rules become simple equations.

Other examples of universal algebras include rings, semigroups, quasigroups, groupoids, magmas, loops, and more. Vector spaces over a fixed field and modules over a fixed ring are also universal algebras. They use addition and scalar multiplication as operations.

Examples of relational algebras include semilattices, lattices, and Boolean algebras.

Basic constructions

In universal algebra, there are three important ways to build new structures from existing ones: homomorphic images, subalgebras, and products.

A homomorphism is a special kind of mapping between two algebraic structures that keeps their operations consistent. A subalgebra is a smaller part of a larger structure that still follows all the same rules. A product combines several structures into one bigger structure by pairing up their elements and performing operations separately on each part.

Some basic theorems

Some important ideas in universal algebra help us see how different parts of math are connected.

The isomorphism theorems show that groups, rings, and modules can sometimes look very similar, even though they are different kinds of structures.

Another big idea is Birkhoff's HSP Theorem. This theorem explains when a group of math structures forms a special type, called a variety. It includes all structures made by changing, shrinking, or combining existing ones.

Motivations and applications

Universal algebra is a part of mathematics that studies different types of algebraic structures. Instead of looking at specific structures like groups or rings, it examines all possible structures and how they connect. This helps mathematicians understand and simplify complex ideas by seeing them in a bigger picture.

Universal algebra has many useful applications. It can be used to study structures like monoids, rings, and lattices. Before universal algebra, important ideas had to be proven separately for each type of structure. With universal algebra, these ideas can be proven once and used for all types of algebraic systems. It also helps solve problems, such as the constraint satisfaction problem (CSP), where the goal is to see if certain conditions can be met within a given structure.

Main article: Constraint satisfaction problem

Generalizations

Further information: Category theory, Operad theory, Partial algebra, and Model theory

Universal algebra can be studied using a special branch of mathematics called category theory. This method helps describe algebraic structures by using categories, which are groups of math rules.

One way to do this is through Lawvere theories or algebraic theories. Another way is through monads. These methods are related and useful in different situations.

Mathematicians also use operad theory, which looks at operations and their rules. This helps connect ideas from different areas of algebra, like rings and vector spaces.

Another area is partial algebra, where operations work only on certain inputs. Model theory combines universal algebra with logic to study math structures in new ways.

History

In Alfred North Whitehead's book A Treatise on Universal Algebra, published in 1898, the term universal algebra meant much the same as it does today. Whitehead said that William Rowan Hamilton and Augustus De Morgan helped start the subject, and James Joseph Sylvester first used the term.

Not many people worked on this subject until the early 1930s. Then, Garrett Birkhoff and Øystein Ore began writing about universal algebras. In the 1940s and 1950s, new ideas in metamathematics and category theory helped the field grow. Since William Lawvere's thesis in 1963, ideas from category theory have been very useful in universal algebra.