23 Types Of Algebra | Curious Desire

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.

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

Aatka Azhar

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.