Outline of algebraic structures

In mathematics, we study many types of algebraic structures. Abstract algebra looks at these structures and their special properties. An algebraic structure usually starts with one or more sets and one or more binary operations or unary operations that follow a list of rules called axioms.

Another part of math, called universal algebra, studies all algebraic structures together. From this view, most structures can be grouped into varieties and quasivarieties based on the rules they follow. Some axiomatic formal systems that don’t fit into these groups are still often counted as algebraic structures.

There are so many algebraic structures today that this article cannot list them all. Sometimes the same structure has different names, or the rules for it might be written in different ways by different writers. This page will mostly cover the structures that most writers agree on.

Study of algebraic structures

Algebraic structures are important in mathematics. When you start learning about them, you first meet structures like groups, vector spaces, and fields. These are groups of things, or sets, that follow certain rules.

As you learn more, you discover more about these structures. Abstract algebra studies the properties of these structures. Universal algebra looks at them in a general way. Category theory shows how different structures, both algebraic and non-algebraic, are related. For example, the fundamental group of a topological space can tell us more about that space.

Types of algebraic structures

In algebra, we study different types of structures using sets and operations. A set is a collection of items, and an operation tells us how to combine these items. One common operation is called a binary operation because it combines two items at a time.

Some basic structures include:

• Set: Just a collection of items with no operations.
• Magma or groupoid: A set with one binary operation.
• Semigroup: A magma where the operation works in a chain (associative).
• Monoid: A semigroup with a special item that doesn’t change things when used in the operation (identity element).
• Group: A monoid where every item has a matching item that brings it back to the identity (inverse elements).

When we have two binary operations, we get structures like:

• Ring: A structure with two operations, often called addition and multiplication, where multiplication follows a rule called the distributive law.
• Field: A special kind of ring where division works for all nonzero elements.

These structures help mathematicians understand patterns and solve problems in many areas.

Algebraic structures with additional non-algebraic structure

Some math structures mix algebraic rules with other ideas. For example, topological vector spaces are vector spaces with a special way to organize points.

Other examples include Lie groups, which are special groups that also have a shape. There are also ordered groups, ordered rings, and ordered fields, where normal math operations work with a way to arrange numbers. Also, Von Neumann algebras are special algebras on a Hilbert space that use a certain topology called the weak operator topology.

Algebraic structures in different disciplines

Algebraic structures are important in many subjects besides mathematics. They are useful in physics, logic, and computer science.

In physics, structures like Lie groups, such as the orthogonal groups and unitary groups, help us understand shapes and turns. Other physics structures include Lie algebras, spaces with special rules, and quaternions.

In mathematical logic, tools like Boolean algebras are very helpful. They work like both patterns and steps. In computer science, structures like Max-plus algebra help us understand how computers work and process information. These examples show how algebraic structures are used in many areas.