Fuzzy Ideals Research Articles

An Application of Generalized Fuzzy Ideals in Ordered Semigroups

Ordered semigroups are algebraic systems consisting of a nonempty set, an associative binary operation, and a partial order compatible with this binary operation. This concept is a generalization of semigroups. One mathematical tool used to study ordered semigroups is the concept of ideals. It turns out that ordered semigroups can be decomposed based on their regularities using various kinds of ideals. The concept of fuzzy sets is one of the mathematical tools used to investigate ordered semigroups, specifically through so-called fuzzy ideals, which are more appropriate than set-theoretical ideals. It is known that the concepts of $(\alpha, \beta)$-fuzzy $(m, n)$-ideals and $(\alpha, \beta)$-fuzzy $n$-interior ideals are generalizations of various types of ideals and many kinds of fuzzy ideals in ordered semigroups. In this paper, we apply the notions of $(\alpha, \beta)$-fuzzy $(m, n)$-ideals and $(\alpha, \beta)$-fuzzy $n$-interior ideals to classify ordered semigroups into classes based on their regularities, using the meaning of characteristic functions.

Characterizations of Regular and Intra-regular Ordered Semigroups by using Generalized Interval Valued Bipolar Fuzzy Quasi-Ideals

In this article, we introduce the concept of a generalized interval-valued bipolar fuzzy quasi-ideal and investigate its properties. We explore the relationship between generalized interval-valued bipolar fuzzy quasi-ideals and generalized interval-valued bipolar fuzzy ideals. Furthermore, we characterize regular and intra-regular ordered semigroups by utilizing generalized interval valued bipolar fuzzy quasi-ideals.

An Innovative Perspective on Bipolar Fuzzy Fantastic Ideals in BCK/BCI-Algebras

In this paper, we propose the concept of $(\in, \in \vee (\varphi^{\divideontimes}, \check{q_{\varphi}}))$-bipolar fuzzy ideals in BCK/BCI-algebras. We show that an $(\in, \in \vee \check{q_{\varphi}})$-bipolar fuzzy ideal is an $(\in, \in \vee (\varphi^{\divideontimes}, \check{q_{\varphi}}))$-bipolar fuzzy ideal. For a BCK/BCI-algebra, it has been shown that an $(\in, \in \vee (\varphi^{\divideontimes}, \check{q_{\varphi}})$-bipolar fuzzy ideal is an $(\in, \in \vee \check{q})$-bipolar fuzzy ideal of $\check{\aleph}$, but not conversely, and then an example is given. We introduce the concept of $(\in, \in \vee (\varphi^{\divideontimes}, \check{q_{\varphi}}))$-bipolar fuzzy fantastic ideals in BCK/BCI-algebras. It has been shown that an $(\in, \in \vee (\varphi^{\divideontimes}, \check{q_{\varphi}}))$-bipolar fuzzy fantastic ideal is an $(\in, \in)$-bipolar fuzzy ideal in BCK/BCI-algebras. Furthermore, the connection between $(\in, \in \vee (\varphi^{\divideontimes}, \check{q_{\varphi}}))$-bipolar fuzzy fantastic ideals and fantastic ideals are established.

Neutrosophic Fuzzy Ideal Open Sets, Ideal Closed Sets, and Their Applications in Geographic Information Systems

Neutrosophic Fuzzy Ideal Open Sets, Ideal Closed Sets, and Their Applications in Geographic Information Systems

Rough and T-Rough Sets Arising from Intuitionistic Fuzzy Ideals in BCK-Algebras

Rough and T-Rough Sets Arising from Intuitionistic Fuzzy Ideals in BCK-Algebras

Normed-Bifuzzy Valued-Ideals of Semigroups

As concerning the views of T norms and T conorms, the intent of article is to define and probe the fuzzy semigroups, fuzzy ideals, fuzzy bi-ideals, bifuzzy subsemigroups, bifuzzy ideals, bifuzzy bi-ideals, fuzzy (1, 2)-ideals and bifuzzy (1, 2)-ideals in any given semigroup. Also, we indicate and study their basic properties of them in completely regular semigroups. Finally, we extend these concepts and so characterize (pre)image of them in semigroup homomorphisms.

Generalizations of prime intuitionistic fuzzy ideals of a lattice

As a generalization of the concepts of an intuitionistic fuzzy prime ideal and a prime intuitionistic fuzzy ideal, the concepts of an intuitionistic fuzzy 2-absorbing ideal and a 2-absorbing intuitionistic fuzzy ideal of a lattice are introduced. Some results on such intuitionistic fuzzy ideals are proved. It is shown that the radical of an intuitionistic fuzzy ideal of L is a 2-absorbing intuitionistic fuzzy ideal if and only if it is a 2-absorbing primary intuitionistic fuzzy ideal of L. We also introduce and study these concepts in the product of lattices.

Exploring Regularities of Ordered Semigroups Through Generalized Fuzzy Ideals

Exploring Regularities of Ordered Semigroups Through Generalized Fuzzy Ideals

Hybrid structure of maximal ideals in near rings

A hybrid structure is an arrangement that makes use of many hierarchical reporting structures and is applied to algebraic structures such as groups and rings. In the discipline of abstract algebra, an ideal of a near-ring is a unique subset of its elements in ring theory. Ideals generalize specific subsets of integers, such as even numbers or multiples of three. Researchers have been using mathematical theories of fuzzy sets in ring theory to explain the uncertainties that emerge in various domains such as art and science, engineering, medical science, and in environment. By fusing soft sets and fuzzy sets, a new mathematical tool that has significant advantages in dealing with uncertain information is provided. Consequently, there is always some discrepancy between reality's haziness and its mathematical model's precision. Hence ring theory has been widely used in many researches but there is some uncertainty in converting the fuzzy sets to a hybrid structure of any algebraic structure. Many approaches were done in groups. Therefore, the Hybrid structure of fuzzy sets in near rings is introduced, in which the fuzzy ideals are converted to hybrid ideals and fuzzy maximal ideals are converted to hybrid maximal ideals. For hybridization, firstly the hybrid structure is established and then sub-near rings and near rings are also determined. Then the hybrid structure of sub-near rings and ideals is introduced. This converts the fuzzy ideals and fuzzy maximal ideals to hybrid ideals and hybrid maximal ideals. The result obtained by the proposed model efficiently solved the uncertainty problems and the effectiveness of the proposed approach shows the best class, mean, worst class, and time complexity.

Generalized roughness of three dimensional (∈,∈∨q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\in ,\\in \\vee q$$\\end{document})-fuzzy ideals in terms of set-valued homomorphism

The objective of this study is to generalize the roughness of a fuzzy set-in three-dimensional structure by introducing ternary multiplication. Many results and theorems of rough fuzzy ideals have been extended from semigroup and semiring, to ternary semiring by introducing the definition of a rough fuzzy subset of ternary semiring. By using the concept of set-valued homomorphism and strong set-valued homomorphism, it is proved generalized lower and upper approximations of (∈,∈∨q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\in , \\in \\vee q)$$\\end{document}-fuzzy ideals (semiprime and prime ideals) of ternary semirings are (∈,∈∨q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\in ,\\in \\vee q)$$\\end{document}-fuzzy ideals (semiprime and prime ideals) respectively.

A New Approach of Complex Fuzzy Ideals in BCK/BCI-Algebras

The concept of complex fuzzy sets, where the unit disk of the complex plane acts as the codomain of the membership function, as an extension of fuzzy sets. The objective of this article is to use complex fuzzy sets in BCK/BCI-algebras. We present the concept of a complex fuzzy subalgebra in a BCK/BCI-algebra and explore their properties. Furthermore, we discuss the modal and level operators of these complex fuzzy subalgebras, highlighting their importance in BCK/BCI-algebras. We study various operations, and the laws of a complex fuzzy system, including union, intersection, complement, simple differences, and bounded differences of complex fuzzy ideals within BCK/BCI-algebras. Finally, we generate a computer programming algorithm that implements our complex fuzzy subalgebras/ideals in BCK/BCI-algebras procedure for ease of lengthy calculations.

Fuzzy ideals and fuzzy filters of bounded semihoops

Fuzzy ideals and fuzzy filters of bounded semihoops

Ideals and Homomorphism Theorems of Fuzzy Associative Algebras

Based on the definitions of fuzzy associative algebras and fuzzy ideals, it is proven that the intersections of fuzzy subalgebras are fuzzy subalgebras, and the intersections of fuzzy ideals are fuzzy ideals. Moreover, we prove that the kernels of fuzzy homomorphisms are fuzzy ideals. Using fuzzy ideals, the quotient structures of fuzzy associative algebras are constructed, their corresponding properties are discussed, and their homomorphism theorems are proven.

(β,α)-magnified Translation of Fuzzy ideals on RG-algebra

In this paper, the concepts of (β,α)-magnified translation of fuzzy RG-subalgebras, (β,α)-magnified translation fuzzy ideals and (β,α)- magnified translation fuzzy q-ideals of RG-algebras are introduced and study several basic properties of them. The following results are obtained and several Theorems and properties are stated and proved.

Generalized interval valued fuzzy ideals in semigroups

Generalized interval valued fuzzy ideals in semigroups

More Results on Intuitionistic Fuzzy Ideals of BE-algebras

This paper explores the intuitionistic fuzzy ideals in BE-algebras and establishes several new results related to their structure. We investigate the fundamental concepts and properties of intuitionistic fuzzy ideals and provide characterizations of an intuitionistic fuzzy ideal in BE-algebras. Our study focuses on examining the fundamental concepts and properties of these ideals and provides characterizations of intuitionistic fuzzy ideals in BE-algebras.

An efficient approach to study multi-polar fuzzy ideals of semirings

The multi polar fuzzy (m-PF) set has an extensive range of implementations in real world problems related to the multi-polar information, multi-index and multi-attributes data. This paper introduces innovative extensions to algebraic structures. We present the definitions and some important results of m-polar fuzzy subsemirings (m-PFSSs), m-polar fuzzy ideals (m-PFIs), m-polar fuzzy generalized bi-ideals (m-PFGBIs), m-polar fuzzy bi-ideals (m-PFBIs) and m-polar fuzzy quasi-ideals (m-PFQIs) in semirings. The main contributions of the paper include the derivation and proof of key theorems that shed light on the algebraic interplay and computational aspects of m-polar fuzzy ideals (m-PFIs), m-polar fuzzy generalized bi-ideals (m-PFGBIs), m-polar fuzzy bi-ideals (m-PFBIs) and m-polar fuzzy quasi-ideals (m-PFQIs) in semirings along with examples. Moreover, this paper deals with several important properties of m-PFIs and characterizes regular and intra-regular semirings by the properties of these ideals.

Retracted: Approximations of Intuitionistic Fuzzy Ideals over Dual Spaces by Soft Binary Relations

Retracted: Approximations of Intuitionistic Fuzzy Ideals over Dual Spaces by Soft Binary Relations

Description of regular and intra-regular ordered semigroups by tripolar fuzzy ideals

Description of regular and intra-regular ordered semigroups by tripolar fuzzy ideals

Retracted: More General Form of Interval-Valued Fuzzy Ideals of BCK/BCI-Algebras

Retracted: More General Form of Interval-Valued Fuzzy Ideals of BCK/BCI-Algebras