Outline of algebraic structures

In mathematics, many types of algebraic structures are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates one or more sets with one or more binary operations or unary operations satisfying a collection of axioms.

Another branch of mathematics known as universal algebra studies algebraic structures in general. From the universal algebra viewpoint, most structures can be divided into varieties and quasivarieties depending on the axioms used. Some axiomatic formal systems that are neither varieties nor quasivarieties, called nonvarieties, are sometimes included among the algebraic structures by tradition.

Algebraic structures are so numerous today that this article will inevitably be incomplete. In addition to this, there are sometimes multiple names for the same structure, and sometimes one name will be defined by disagreeing axioms by different authors. Most structures appearing on this page will be common ones which most authors agree on.

Study of algebraic structures

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

As you go further, you learn more about these structures. Abstract algebra looks at the properties of these structures. Universal algebra studies them in a more general way. Category theory looks at how different structures, both algebraic and non-algebraic, relate to each other. 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 just 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 mathematical structures combine algebraic properties with other types of structures. For example, topological vector spaces are vector spaces that also have a special way of organizing points called a topology.

Other examples include Lie groups, which are special kinds of groups that also have a geometric shape. There are also ordered groups, ordered rings, and ordered fields, where the usual algebraic operations work together with a way to arrange the numbers. Additionally, Von Neumann algebras are special types of algebras on a Hilbert space that use a particular topology known as the weak operator topology.

Algebraic structures in different disciplines

Algebraic structures are not only important in mathematics but also have applications in other fields like physics, logic, and computer science. In physics, for example, structures like Lie groups, including the orthogonal groups and unitary groups, help describe symmetries and rotations. Other physics-related structures include Lie algebras, inner product spaces, and quaternions.

In mathematical logic, tools like Boolean algebras, which act as both rings and lattices, are essential. Computer science uses structures such as Max-plus algebra to model computations and processes. These examples show how algebraic structures support many areas beyond pure math.