Algebra is a branch of mathematics that consists of rules and procedures for manipulating numbers. The first person to use the symbol x to represent an unknown number was Rene Descartes in 1637. He also wrote that “to find the value of ‘x’ for any given starting point. “Algebra has changed over time.

Are you wondering about the types of algebra and their functionalities? So, this article is for you as it will discuss 23 different types of algebra.

Table of Contents

**23 Types Of Algebra That You Might Not Know**

Algebra is used across many disciplines including science, engineering, business, economics, etc. Babylonians made arithmetic and algebraic notation very popular and widely used. Today, many types of algebra focus on different concepts depending on the intended applications.

For example, linear algebra deals with the study of vector spaces while abstract algebra is more theoretical. Following are the 23 types of algebra.

### 1. Abstract Algebra

Abstract algebra is the study of algebraic structures. In particular, it studies groups, rings, fields, and vector spaces intending to find general methods for solving equations. Abstract algebra provides a powerful toolset to study real-world problems. For example, abstract algebras have been used in physics to solve difficult problems such as classifying all possible symmetries of a particle.

### Fields

Abstract algebra has many subdisciplines. Two of the most popular are ring theory and field theory. They are considered different areas because they focus on different concepts.

- Ring
- Group
- Vector Space

### 2. Linear Algebra

Linear algebra is the study of vector spaces and linear transformations. Vector spaces are mathematical structures in which you can add or subtract two vectors. Linear transformations are differentiable maps that send a vector in one vector space to another vector space, these maps must be additive inverses or have an inverse that is also a linear transformation. The first person to use the term “linear algebra” was Peano in 1888.

### Fields

Linear algebra has many fields, for example

- Differential Linear Algebra
- Matrix Algebra
- Group Linear Algebra
- Finite-Dimensional Vector Spaces
- Geometric Linear Algebra
- Tensor Analysis With Applications In Mechanics And Physics
- Linear Transformations in a vector space

### 3. Geometry Algebra

The main purpose of geometry is to answer the question “How are points, lines, and planes related in space?” Geometry has many different branches including Euclidean geometry, projective geometry, and finite geometry.

They all have different approaches but all of them attempt to get a better understanding of space.

### Fields

Geometry algebra is a branch that focuses on spatial objects and their properties. It makes use of both Euclidean geometry and coordinate geometry giving it an advantage over other branches, for example, the ability to represent the location of an object in space using coordinates. Some of the most important fields of geometry algebra are

- Affine Geometry
- Differential Geometry
- Differential Forms In Algebraic Topology
- Elementary Differential Geometry
- The Clifford Algebra-Applications And Examples Volume 1
- Geometric Models For Noncommutative Algebras
- Geometry, Topology and Physics A – Z: Geometric Methods In Theoretical Physics

### 4. Boolean Algebra

The purpose of Boolean algebra is to develop a system that has the properties of set theory and logic. It was derived from propositional calculus which is used in mathematics, philosophy, linguistics, and computer science.

This branch discusses how variables can be combined to form expressions. For example, the expression (a+b)∨(a-b) would be considered to be equivalent to (a+b)∧(a-b). The first person who used Boolean algebra was George Boole, an English mathematician.

### Fields

Boolean algebra has many fields including

- Combinatory Logic
- Logic Algebra
- Elementary Boolean Algebra
- Basic Properties Of Boolean Functions
- Algebras And Operator Theory Volume I General Structures And Their Applications

### 5. Topology Algebra

Topology is the study of properties that are preserved under continuous deformations of objects. This is a branch of math that uses algebra to study topological spaces. A very big part of topology is a thing called homotopy, which is a way of measuring the “distance” between two continuous functions from one space to another.

In topology, we can take some data and send it on a roller coaster ride, which will impact it in a way that we can’t imagine. This is a huge field of abstract mathematics with applications to computer science and economics. It’s an important part of modern physics too, especially quantum computing.

### Fields

The following are just some of the possible Fields of Topology algebra

- Differential-Cocycles, Lie Algebras, And Differential Equations
- Operator Algebras and Quantum Statistical Mechanics
- Homotopy Theory and Univalent Functions
- Homotopical Algebra and its Applications

### 6. Logic Algebra

This branch studies the relationship between sets and logic systems. Logic algebras can represent information about sets, logic statements, and operations. It’s used to prove theorems in mathematics, geometry, physics, and computer science.

The first person who used this type of algebra was George Boole, an English mathematician. The purpose of his work was to develop a system that had the properties of set theory and logic.

### Fields

Logic algebra is composed of many different fields including

- Boolean fields
- Categorical Logic
- Computability Theory And Complexity
- Temporal Logical Formalization Of Programming Languages
- Logic And Algebra Of Specification
- Quantum Logic And Nonclassical Logics
- Algebraic Theory Of Computation, A

### 7. Fuzzy Algebra

This is a branch of abstract algebra that deals with numbers whose values are on some spectrum between true or false, such as “I am feeling slightly hungry” or “I am a little embarrassed”. It can be used to represent data that has multiple states, such as the speed of a car being slightly over the speed limit.

This type of algebra uses fuzzy sets which are multisets that have a membership grade between 0 and 1. The first person to use these types of concepts was Lotfi Zadeh, who is best known for creating fuzzy logic.

### Fields

Many different fields relate to fuzzy algebra, including

- Cognitive Science And Mathematical Modelling
- Fuzzy Logic And Its Applications
- Approximate Reasoning In Intelligent Systems Using Fuzzy Sets

### 8. Advanced Algebra

This branch of abstract algebra is mostly used in mathematics, physics, and engineering. It studies rings, modules over a ring, vector spaces, associative algebras, and Lie algebras. A ring is an additive group with unity for which you can multiply elements together to get other elements in the group.

A module is a ring that has some direct summand or factor that can be considered to be its subring. An associative algebra is a vector space with an additional multiplication operation that makes it into a ring, which allows you to do matrix operations without changing the result of the calculation. Vector spaces have infinite dimensions, which means they, abstract groups. To do so, you must use an associative algebra that is also unital.

### Fields

These fields all relate to advanced algebra. They include

- Lie Theory And Lie Algebras
- Integrable Systems, Representations, and Quantum Field Theories
- Adjoint Representations and Superalgebras
- Groups and Their Generalizations: A Unified Approach
- Representation Theory of Finite Groups and Associative Algebras

### 9. Elementary Algebra

This is the study of real numbers, complex numbers, quaternions, and polynomials. It also deals with linear equations, exponential functions, and logarithms. A good way to think about elementary algebra is to see it as arithmetic with letters. This field is also sometimes called “straight-line algebra”.

### Fields

Many different subfields relate to elementary algebra, including

- Computer Arithmetic
- Trigonometry
- Number Theory And Field Theory
- Algebraic Number Fields And Function Fields

### 10. Knot Algebra

Knot theory is the study of mathematical knots, which are shapes that interlink in a strand-like fashion. Curly brace notation is used to represent this type of structure, and it looks like this: { }. You can take any set of knots and consider their knot structures to be an

This branch of abstract algebra deals with the mathematical study of knots, which are not to be confused with knotted shoelaces or double-helical DNA. It’s an isolated branch that doesn’t have much cross-over with other areas of abstract algebra.

### 11. Operator Algebras

Operator algebras are particularly important in quantum mechanics, where they are used to represent observables. This type of algebra is also well-known in the study of infinite-dimensional algebras.

### Fields

These fields all relate to operator algebras. They include

- Algebraic Quantum Theory
- Free Resolvents In Commutative Rings With Applications

### 12. Categorical Algebra

This is a branch of abstract algebra that deals with structures and how they relate. When you’re looking at categorical algebras, it’s important to remember that you can define them in many different ways.

The usual way categorifies define structures is through homomorphism, which means something similar to “similar”. There are other ways to define structures, however.

### Fields

These fields all relate to categorical algebra. They include

- Semantics Of Computational Structures
- Univalent Foundations
- Foundation Of Quantum Gravity
- Geometric Langland’s Program (GLP)

### 13. Algebras Of Sets And Classes

These types of abstract algebra study sets and classes, which are different types of structures that represent sets. Set theory is the mathematical rule used to describe collections of things. These are fundamental in areas such as math, computer science, linguistics, and information science.

The first person who worked in this field was Giuseppe Peano, who also created the concept of a natural number. Classes are ways to describe objects and can be used as part of set theory or topology algebra.

### Fields

These fields all relate to algebras of sets and classes. They include

- Set Theory
- Topology Algebra
- Knot Theory
- Boolean Algebra
- Complex Geometry And Topology (CGT)
- Geometric Topology

### 14. Universal Algebra

This branch is about creating systems where you take some types of elements and use them to build new series of numbers, sets, structures, and mappings. You create structures based on some types of elements; this is what universal algebra deals with. The first person to use these concepts was Richard Dedekind.

### Fields

These fields all relate to universal algebra. They include

- Algebras And Rings With A Unital Operation
- Boolean Algebras And Boolean Rings
- Brouwer Fixed-Point Theorem
- Complex Structures On An Algebraic Variety
- Derived Categories
- Finitely Generated Algebras
- Matrix Algebras And Matrix Theory
- Nonassociative Rings And Algebras
- Representation Theory Of Finite Groups And Associative

### 15. Weak Algebra

This branch of abstract algebra deals with the study of some equivalence relation that has three different properties: it’s symmetric, transitive, and has the reflexive property. Weak algebra is sometimes called abstract theory since it deals with objects that are not necessarily sets or classes. It’s used to describe groups of numbers that may have some equivalences, even if they’re not similar themselves.

### Fields

These fields all relate to weak algebra. They include

- Categorical Group Theory
- Category Of Sets And Category Of Graphs
- Group Theory In Categories And Groupoids
- Non-Associative Algebras
- Ordered Sets
- Relations On A Set Or On A-Class
- Tensor Algebra
- Universal Algebra

### 16. Tensor Algebra

This branch deals with tensors which are a generalization of the multilinear map that have coefficients in some ring or field. They can be used to describe geometric objects, such as vectors. The basis for this type of algebra was developed by Hermann Grassmann, although he never named it tensor algebra. He used it to describe geometric objects and linear algebra.

### Fields

These fields all relate to tensor algebras. They include

- Dimension Theory
- Lie Groups And Lie Algebras
- Matrix Transformations (Mt)
- Tensor Categories And Tensor Factors

### 17. Finitely Generated Algebra

This branch deals with algebras that have a finite basis, which is the term used to describe the smallest number of elements required for some set or class. The first person who worked on this concept was John von Neumann.

This type of algebra uses the concept of generating sets, which are ways to describe an infinite series in very little space.

### Fields

These fields all relate to finitely generated algebras. They include

- Algebras Of Functions
- Commutative Rings And Their Applications
- Compact Operators And Embeddings
- Contributions To Algebra And Geometry
- Differentiable Manifolds
- Finitely Generated Modules And Fg-Modules
- Function Spaces And Function Zones
- Group Rings Of Finite Groups

### 18. Graph Algebra

This branch deals with graph theory, which is the study of structures that are made up of nodes and edges. They can be used to represent different types of data, such as social networks. One of the most important parts of graph theory is incidence algebra, which studies the relationship between points and lines.

### Fields

These fields all relate to graph algebras. They include

- Boolean Functions And Applications
- Categorical Combinatorics
- Commuting Biquandles
- Combinatorial Algebra
- Complexes Of Graphs
- Finite Geometries
- Graphical Sequences And Their Applications
- Graphs In Computer Science
- Graphs In Mathematical Systems

### 19. Matrix Algebras

This branch is a form of linear algebra, which is the study of vector spaces that have a specific set of properties. Matrix algebras deal with matrices, which are two-dimensional arrays that are used for solving systems of linear equations. They’re also used for other purposes.

The person who first studied matrix algebras was Emmy Noether, although she never named them since matrices were out of fashion at that time.

### Fields

These fields all relate to matrix algebras. They include

- Algebras Of Operators On Hilbert Space
- Bilinear Forms And Applications
- Enumeration Methods For Combinatorial Problems
- Graphs And Matrices (Gm)
- Linear Algebra, Linear Transformations (Lt)
- Matrix Spaces Over The Reals And Vector Spaces

### 20. Commutative Algebra

This branch of algebra is related to commutative rings, which are fields that have the property where multiplication is commutative. Commutative algebras are used in combinatorics since they are an important part of enumeration. People who study this type of algebra often work on generalizations of these types of structures. They’re also used to study polynomial rings.

### Fields

These fields all relate to commutative algebras. They include

- Algebraic Combinatorics
- Algebraic Number Theory
- Binary Polynomials And Polynomial Rings
- Coding Theory
- Enumeration Methods For Structural Design Problems
- Free Resolvents In Commutative Algebras With Applications
- Group-Like Modules Over Commutative Rings
- Group Theory And Combinatorics
- Partial Orders In Algebraic Structures

### 21. Computational Algebra

This branch uses computation to do algebra, creating algorithms that solve different problems. It’s also called “computer algebras” since it deals with computations that are done on different structures like sets or classes.

The first person who worked on this type of algebra was Nathaniel Rochester, who created the first programming language called Auto code in 1950.

### Fields

These fields all relate to computational algebras. They include

- Coding Theory
- Computer Algebra
- Decision Problems In Computational Algebras
- Finitely Generated Modules And Computer Generation Of Structures
- Graphical Models And Image Processing Using Monoids And Semigroups

### 22. Homological Algebra

This branch is about studying rings and modules, which are different types of algebraic structures that can be used to solve some problems. It’s typically considered an extension of commutative algebra.

Homological algebras are based on homology, which is the study of different classes that are uniquely determined by some data. It was developed during the 1960s.

### Fields

These fields all relate to homological algebras. They include

- Combinatorial Homology Theory
- Dynamics On Homogeneous Spaces And Cohomology
- Free Resolvents In Commutative Rings With Applications

### 23. Ternary Algebra

This branch is about studying different types of numbers that can be represented by a ternary numeral system. Ternary algebra is also called “balanced algebras”, since they have more than two summands, unlike common algebra which has only two elements as a result.

A well-known example of this type is noncommutative geometry, which is a branch in quantum field theory.

### Fields

These fields all relate to ternary algebras. They include

- Noncommutative Geometry
- Physics With Novel Ternary Quantum Numbers
- Operator algebras

## Types Of Algebra – Conclusion

As you can see, there are many types of algebra, and they’re used in different areas. However, these branches tend to overlap and influence each other at the same time. This makes it difficult to study them separately.

What we can say for sure is that algebra allows us to solve problems that we wouldn’t be able to do otherwise, and its importance is increasing in modern sciences. This makes studying the different branches of algebra very important for students who want to be able to understand discoveries and theories, or even contribute new ones.

#### References

- https://byjus.com/maths/algebra/
- https://www.vedantu.com/maths/algebra
- https://www.cuemath.com/algebra/

Hello, I am Aatka Azhar. I have done bachelor’s degree in Computer Science. Writing articles is my passion through which I explore so many things. So, I am using my passion by working here and it will be the source of knowledge for the readers.