Algebraic structure

In mathematics, an algebraic structure is a way of organizing numbers and symbols so we can perform operations like addition and multiplication in a clear and consistent way. It starts with a collection of items, called a set, and defines rules for how these items can be combined. These rules are known as identities or axioms, which help ensure that the operations behave in predictable ways.

Algebraic structures can build on one another. For example, a vector space combines another structure called a field with special rules for multiplying these elements. This idea is a key part of abstract algebra, the branch of math that studies these structures.

The study of algebraic structures is formalized in areas like universal algebra and category theory, which look at patterns and relationships between different kinds of structures. These tools help mathematicians understand the deep connections that tie many areas of math together.

Introduction

Addition and multiplication are typical examples of operations that combine two elements of a set to produce a third element of the same set. These operations follow certain rules, like the associative law, where a + (b + c) equals (a + b) + c, and the commutative law, where a + b equals b + a. Many mathematical systems have operations that follow some, but not all, of these usual arithmetic rules.

Sets with one or more operations that follow specific rules are called algebraic structures. When we encounter a new problem that uses the same rules as an algebraic structure, we can use what we already know about that structure to solve the problem. Algebraic structures can include operations that combine more than two elements, operations that use just one element, or even operations that use no elements at all.

Common axioms

Algebraic structures use special rules, called axioms, to describe how their operations work. One common type of axiom is an identity, which is a simple equation that must always be true. For example, an operation is commutative if swapping the order of the elements does not change the result, like addition: 3 + 4 is the same as 4 + 3. Another example is associativity, where grouping does not matter: (1 + 2) + 3 equals 1 + (2 + 3).

Some axioms include an existential clause, meaning they state that a certain element must exist. For instance, an operation might need an identity element, like 0 in addition, where adding zero does not change the number: 5 + 0 = 5. Other structures might require elements to have an inverse, such as −5 for the number 5, because 5 + (−5) = 0.

Common algebraic structures

Main article: Outline of algebraic structures § Types of algebraic structures

Algebraic structures are ways to organize numbers and operations in mathematics. Some simple structures have just one set of numbers with no special operations. Others, like groups, have one operation that works in specific ways. For example, a group is a set where you can combine any two elements using this operation, and every element has a matching "inverse" element.

More complex structures, like rings, have two operations — often called addition and multiplication — where one operation distributes over the other. Fields are special kinds of rings where you can divide by any non-zero number. There are also structures that involve two sets with operations, such as vector spaces, where one set acts on another in a defined way.

Hybrid structures

Algebraic structures can have extra features that are not algebraic, such as a way to organize them or a special kind of arrangement. These extra features need to work well with the algebraic structure.

Some examples include:

• A topological group: a group with a special arrangement that works with its group operation.
• A Lie group: a topological group with a smooth structure.
• Ordered groups, ordered rings and ordered fields: structures with a way to compare elements.
• An Archimedean group: a group where a special property called the Archimedean property is true.
• A topological vector space: a vector space with a special arrangement.
• A normed vector space: a vector space with a measure of size.
• A Hilbert space: a space with a way to measure angles and distances.
• A Von Neumann algebra: a special type of algebra on a Hilbert space.

Universal algebra

Main article: Universal algebra

Algebraic structures are defined by rules called axioms. Universal algebra studies these structures in a general way. Some structures follow rules that are simple equations, called identities. When all the rules are identities, the structure is called a variety.

For example, groups are a type of algebraic structure. They have special rules, like how to combine elements and how to reverse those combinations. These rules help create a consistent system. Other structures, like fields, have more complex rules and do not always fit into the simple identity rules. This makes them more challenging to study.

Category theory

Category theory is a way to study algebraic structures. A category is a group of objects with special links called morphisms between them. Every algebraic structure has its own type of link, called a homomorphism, which works with the structure's operations. For example, the category of groups includes all groups as objects and special links called group homomorphisms as morphisms.

Category theory includes many ideas that help describe algebraic structures, such as algebraic categories, essentially algebraic categories, presentable categories, locally presentable categories, monadic functors and categories, and universal properties.

Different meanings of "structure"

The word "structure" in math can sometimes mean just the operations used on a set, not the set itself. For example, saying "we have a ring structure on set A" means we’ve added certain operations to set A, making it a ring. Another example is the group (Z, +), which is the set of integers Z with the operation of addition (+). This shows how operations give structure to a set.

Main article: Ring