Universal algebra

Universal algebra is a branch of mathematics that studies algebraic structures in general. Instead of looking at specific types like groups or rings, which have their own theories, universal algebra looks at the possible types of algebraic structures and how they relate to each other. This field helps mathematicians understand the common patterns and properties shared by different algebraic systems. By studying these structures broadly, universal algebra provides tools and ideas that are useful in 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 along with some operations you can perform on the elements of that set. These operations can combine elements in different ways.

Arity

Main article: Arity

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

Varieties

Main article: Variety (universal algebra)

Not to be confused with Algebraic variety.

A collection of algebraic structures defined by equations is called a variety or equational class. This means we only use equations to describe these structures, without using concepts like "for all" or "there exists" beyond simple equations, or using relations other than equality.

The study of these equational classes is a special part of model theory, focusing on structures with operations only, like functions, and using only equations to describe them. For example, ordered groups wouldn't fit here because they use an ordering relation, 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 benefit of this approach is that these structures can be studied in any category that has finite products. For example, a topological group is just a group within the category of topological spaces.

Examples

Most common algebraic systems in math are examples of varieties, though their usual definitions might seem different because they often use concepts 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 slightly 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 without needing "for all" or "there exists" beyond the outermost universal quantifiers.

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, with 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 understand how different mathematical structures relate to each other. The isomorphism theorems show how groups, rings, and modules can look very similar in certain ways even though they are different types of structures.

Another key idea is Birkhoff's HSP Theorem. This theorem tells us that a group of algebraic structures forms a special category, called a variety, exactly when it includes all structures that come from changing parts of existing structures, taking smaller parts of structures, and combining many structures together.

Motivations and applications

Universal algebra is a branch of mathematics that studies algebraic structures in a general way. Instead of focusing on specific types of structures like groups or rings, it looks at the possible types of structures and how they relate to each other. This approach helps mathematicians understand and simplify complex ideas by placing them in a broader context.

Universal algebra has many useful applications. It can be used to study structures like monoids, rings, and lattices. Before universal algebra, important theorems had to be proven separately for each type of structure. With universal algebra, these theorems can be proven once and apply to all types of algebraic systems. It also helps in solving computational problems, such as the constraint satisfaction problem (CSP), where the goal is to determine 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 also be studied using a special branch of mathematics called category theory. This method helps describe algebraic structures by using categories, which are like organized groups of mathematical rules. One way to do this is through something called Lawvere theories or algebraic theories, and another way is through monads. These methods are closely related and useful in different situations.

Recently, mathematicians have also used operad theory, which focuses on operations and their rules in a specific way. This helps in combining ideas from different areas of algebra, like rings and vector spaces. Another area is partial algebra, where operations can sometimes only work on certain inputs. Model theory combines universal algebra with logic to study mathematical structures in a new way.

History

In Alfred North Whitehead's book A Treatise on Universal Algebra, published in 1898, the term universal algebra had essentially the same meaning that it has today. Whitehead credited William Rowan Hamilton and Augustus De Morgan as originators of the subject matter, and James Joseph Sylvester with coining the term itself.

Work on the subject was minimal until the early 1930s, when Garrett Birkhoff and Øystein Ore began publishing on universal algebras. Developments in metamathematics and category theory in the 1940s and 1950s furthered the field. Starting with William Lawvere's thesis in 1963, techniques from category theory have become important in universal algebra.