INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 07,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 21
IDEMPOTENT OPERATIONS AND THEIR APPLICATIONS
Dilnoza Shakirjonovna Djurayeva
Lecturer, Department of “Algebra and Geometry”,
Termez State University
Tel.: + 998 97 343 20 35.
E-mail: dilnozajuraeva@mail.ru
Dilnoza Shunqorbek qizi Kuldasheva
2nd year student, Mathematics,
Termez State University
Abstract:
The article studies idempotent operations and their applications.
Key words and new concepts:
Idempotent operation, idempotent function, idempotent matrix,
linear algebra, special properties
Introduction:
Before explaining the discussion, let me briefly touch on what idempotent action
is and its history, and then give some information about the scientific development in our
country.
Scientific development in our country has been taking place at a serious pace in recent years.
The government is paying great attention to the development of science and technology,
implementing new initiatives and programs to develop scientific research.
Firstly, state support for scientific research and innovation has increased in Uzbekistan. New
scientific institutes and research centers are being established and their activities are being
modernized. Also, programs are being implemented in higher educational institutions aimed at
increasing the level of scientific research.
Secondly, international cooperation is being strengthened. The scientific community of
Uzbekistan is actively participating in international scientific conferences, symposiums and
projects. This creates an opportunity to exchange experience and study new technologies
Thirdly, the results of scientific research are used in economics, agriculture, medicine and other
areas. They are important for developing innovative solutions and new technologies, supporting
economic growth.
Also, many initiatives are being implemented to attract young people to scientific activities.
The goal is to increase their scientific potential by supporting students with scientific research,
grants or scholarships.
Idempotent function and its history
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 07,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 22
A function f(x) is called an idempotent function if it satisfies the following condition:
f(f(x)) = f(x) for any x.
Idempotent functions are a concept that has existed in mathematics for a long time, and they are
important in the theories of functions and operators. The main feature of an idempotent function
is that if you apply an operation several times, the result does not change. That is, if A is an
idempotent action, then A(A(x)) = A(x) is true for any x.
The history of idempotent actions reflects an interesting development in mathematics. These
concepts were formed mainly in the framework of algebra and operator theory.
Objective:
To work on examples of idempotent operations and their applications
Idempotent operations are often used in computer science, in particular in databases and
algorithms.
- They provide stable and consistent operation processes with respect to data changes.
Research object:
To acquire detailed knowledge and skills about idempotent operations among
teachers and students, and to apply it in practice.
An idempotent operation is a property of a mathematical function that, when you apply it once,
the result does not change. Mathematically, if A is an idempotent operation, then the equality
A(A(x)) = A(x) is satisfied. This means that the result does not change when the operation is
performed multiple times.
The main theorem of idempotence: If A is an operation and A^2 = A, then A is an idempotent
operation. This theorem is valid for any operation, including matrices, operators, and many
other structures.
Idempotent function: A function f(x) is idempotent if f(f(x)) = f(x).
Idempotent matrix: A matrix A is idempotent if A^2 = A. Such matrices are used in geometry
as projection operators.
Idempotent operations are important batch operations. For example, performing the same
operation multiple times to update data gives a stable result.
Linear algebra: Projection operators are described as idempotent matrices, and these
transformations represent geometric concepts.
Examples:
1. Idempotent function: The function max(a, b) is a good example, since max(max(a, b)) =
max(a, b).
2. An example of an idempotent matrix is the identity matrix E.
3. Idempotent operation: IIaII=IaI
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 07,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 23
Idempotent operations ensure the stability of software functions, for example, when used in
databases. Even if you perform an operation several times, the result does not change, which
reduces errors and increases the reliability of the system. Idempotent matrices are also used in
geometry to perform projections of objects, which allows you to simplify complex shapes.
These two aspects make idempotent operations very valuable in the field of international
scientific research and technology.
Summary. Idempotent operations, functions, and matrices are important in mathematical
theories. They are used in many areas, in particular in the fields of databases and programming.
The concept of idempotence plays a key role in the development and application of
mathematical thought. Thus, idempotent operations and their theories are important in
mathematics, modern science, and practice. They appear as exemplary mathematical functions
and provide efficient solutions in many areas.
References:
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