Fuzzy Subalgebras Research Articles

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.

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.

Subalgebras of Sheffer Stroke BCK-Algebras Illuminated by the New Fuzzy Set Environment

For the purpose of dealing with the subalgebras of Sheer stroke BCK-algebras associated with the [Formula: see text]-fuzzy set, the concept of [Formula: see text]-fuzzy subalgebras in Sheffer stroke BCK-algebras is introduced and the properties involved are investigated. Conditions for creating a [Formula: see text] fuzzy subalgebra from a given fuzzy are explored. The relationship between fuzzy subalgebra and the [Formula: see text] fuzzy subalgebra is discussed, and conditions are given for the [Formula: see text]-fuzzy subalgebra to be a fuzzy subalgebra. The task of creating a new [Formula: see text]-fuzzy subalgebra is established using the given fuzzy subalgebra. Two types of level sets are created and the conditions under which they become subalgebras are explored.

Multipolar neutrosophic subalgebras/ideals of UP-algebras

The notion of neutrosophic m-polar fuzzy sets is much wider than the notion of m-polar fuzzy sets. In this paper, we apply the theory of neutrosophic m-polar fuzzy set on UP-algebras. We introduce the concepts of neutrosophic m-polar fuzzy subalgebras, neutrosophic m-polar fuzzy ideals and neutrosophic m-polar fuzzy strong ideals and some essential properties are discussed. We characterize neutrosophic m-polar fuzzy subalgebras in terms of fuzzy subalgebras and subalgebras of UP-algebras.

Łukasiewicz Anti Fuzzy Subalgebras of BCK/BCI-Algebras

The subalgebra of BCK/BCI-algebra using Łukasiewicz anti fuzzy set introduced by Jun is studied in this article. The concept of Łukasiewicz anti fuzzy subalgebra of a BCK/BCI-algebra is introduced, and several properties are investigated. The relationship between anti fuzzy subalgebra and Łukasiewicz anti fuzzy subalgebra is given, and characterization of a Łukasiewicz anti fuzzy subalgebra is discussed. Conditions are found in which a Lukasiewicz anti fuzzy set is a Lukasiewicz anti fuzzy subalgebra Finally, conditions under which ⋖-subset, ϒ-subset, and anti-subset become subalgebra are explored.

M-Polar Fuzzy B-ideal of B-algebra

B-algebra is an algebraic structure related to BCI/BCK-algebra. Many researchers have studied fuzzy B-ideal on B-algebra, m-polar fuzzy set on BCI-algebra and B-algebra, m-polar fuzzy subalgebra on BCI-algebra and B-algebra, m-polar fuzzy ideal on BCI-algebra, m-polar (∈,∈)-fuzzy p-ideal on BCI-algebra, m-polar (∈,∈)-fuzzy q-ideal on BCI-algebra, and m-polar (∈,∈)-fuzzy a-ideal on BCI-algebra. We build a new structure, namely m-polar (∈,∈)-fuzzy B-ideal on B-algebra. This research aims to extend the knowledge of m-polar fuzzy sets, which can be combined with other algebraic structures, besides BCI-algebra. In this study, we investigate and describe the properties of m-polar (∈,∈)-fuzzy B-ideal of B-algebra. We also investigate the connection among m-polar (∈,∈)-fuzzy B-ideal, m-polar fuzzy subalgebra, and m-polar fuzzy ideal. We serve a condition that causes an m-polar fuzzy ideal to become an m-polar (∈,∈)-fuzzy B-ideal. We also serve expansion properties of an m-polar (∈,∈)-fuzzy B-ideal. Futhermore, examples showing the modification of π_i formula are added. The properties of m-polar (∈,∈)-fuzzy B-ideal of B-algebra are obtained by combining and modifying the properties of m-polar (∈,∈)-fuzzy p-ideal, m-polar (∈,∈)-fuzzy q-ideal, and m-polar (∈,∈)-fuzzy a-ideal of BCI-algebra

Ideals of BCK-algebras and BCI-algebras Based on a New Form of Fuzzy Set

Ideals in BCK/BCI algebra based on $Y_J^{\varepsilon}$-fuzzy sets are studied. The fundamental properties of the level set of $Y_J^{\varepsilon}$-fuzzy sets are investigate first. The concept of (closed) $Y_J^{\varepsilon}$-fuzzy ideals in BCK/BCI-algebras is introduces, and several properties are investigated. The relationship between $Y_J^{\varepsilon}$-fuzzy ideal and $Y_J^{\varepsilon}$-fuzzy subalgebra are discussed, and also the relationship between $Y_J^{\varepsilon}$-fuzzy ideal and fuzzy ideal is identified. The characterization of (closed) $Y_J^{\varepsilon}$-fuzzy ideal using the Y-level set is established. The necessary and sufficient conditions for $Y_J^{\varepsilon}$-fuzzy ideal to be closed is explored, and conditions for $Y_J^{\varepsilon}$-fuzzy subalgebra to be $Y_J^{\varepsilon}$-fuzzy ideal are provided.

Intuitionistic Fuzzy Ordered Subalgebras in Ordered BCI-algebras

In this paper, we apply the concept of an intuitionistic fuzzy set to ordered subalgebras in ordered BCI-algebras in the sense of intuitionistic fuzzy point. We introduce the notion of an intuitionistic fuzzy (ordered) subalgebra in ordered BCI-algebras, and investigate some related properties. We provide relations between an intuitionistic fuzzy ordered subalgebra and an intuitionistic fuzzy subalgebra. We give characterizations of an intuitionistic fuzzy (ordered) subalgebra. Finally, we provide relations between a $q_{(t,s)}$-level set of intuitionistic fuzzy set and an intuitionistic fuzzy ordered subalgebra.

On Complete Intuitionistic Fuzzy Roh-Ideals in Roh-Algebras

In this paper, we shall investigate and study some kinds of ideals in an intuitionistic fuzzy setting, they are called complete intuitionistic fuzzy subalgebra, complete intuitionistic fuzzy ideal, and complete intuitionistic fuzzy ideal. In this study, we have also proposed some hypotheses to explain some of the relationships between these kinds of intuitionistic fuzzy ideals.

Bipolar complex fuzzy subalgebras and ideals of BCK/BCI-algebras

The conception of the bipolar complex fuzzy set (BCFS) is one of the fundamental and significant modifications of the fuzzy set (FS) to tackle the tricky and awkward information. BCFS has a rich and wider structure and has been utilized in various fields. In this article, we introduce the concept of bipolar complex fuzzy (BCF) subalgebras (BCFSAs), BCF ideals (BCFIs) of a BCK/BCI-algebra along with certain properties. Further, we investigate the relations between BCFSA and a BCFI and a necessary condition for BCFSA to be a BCFI. We also investigate characterizations of BCFI. Moreover, we introduce the notion of equivalence relations on the collection of all BCFIs of BCK/BCI algebra and the associated properties of equivalence relations.

Fuzzy Subalgebras and Ideals With Thresholds of Hilbert Algebras

The concepts of fuzzy subalgebras and ideals with thresholds of Hilbert algebras are presented, some of their features are explained, and their extensions are demonstrated using the theory of fuzzy sets as a foundation. We also talk about the connections between fuzzy subalgebras (ideals) with thresholds and their level subsets. The homomorphic images and inverse images of fuzzy subalgebras and ideals with thresholds in Hilbert algebras are also studied and some related properties are investigated.

Hesitant Fuzzy Structures on Sheffer Stroke BCK-Algebras

The main objective of the study is to introduce a hesitant fuzzy structures on Sheffer stroke BCK-algebras related to their subsets (subalgebras as possible as). Then it is proved that every hesitant fuzzy ideal of a Sheffer stroke BCK-algebra related to the subset is the hesitant fuzzy subalgebra. By defining a hesitant fuzzy maximal ideal in this algebra, relationships between aforementioned structures, subalgebras and ideals on Sheffer stroke BCK-algebras are shown in detail. Finally, it is illustrated that a subset of a Sheffer stroke BCK-algebra defined by a certain element and a hesitant fuzzy (maximal) ideal on the algebra is a (maximal) ideal but the inverse is usually not true.

Anti-hesitant Fuzzy Subalgebras, Ideals and Deductive Systems of Hilbert Algebras

The Hilbert algebra, one of several algebraic structures, was first described by Diego in 1966 [7] and has since been extensively studied by other mathematicians. Torra [18] was the first to suggest the idea of hesitant fuzzy sets (HFSs) in 2010, which is a generalization of the fuzzy sets defined by Zadeh [20] in 1965 as a function from a reference set to a power set of the unit interval. The significance of the ideas of hesitant fuzzy subalgebras, ideals, and filters in the study of the different logical algebras aroused our interest in applying these concepts to Hilbert algebras. In this paper, the concepts of HFSs to subalgebras (SAs), ideals (IDs), and deductive systems (DSs) of Hilbert algebras are introduced in terms of anti-types. We call them anti-hesitant fuzzy subalgebras (AHFSAs), anti-hesitant fuzzy ideals (AHFIDs), and anti-hesitant fuzzy deductive systems (AHFDSs). The relationships between AHFSAs, AHFIDs, and AHFDSs and their lower and strong level subsets are provided. As a result of the study, we found their generalization as follows: every AHFID of a Hilbert algebra Ω is an AHFSA and an AHFDS of Ω. We also study and find the conditions for the complement of an HFS to be an AHFSA, an AHFID, and an AHFDS. In addition, the relationships between the complements of AHFSAs, AHFIDs, and AHFDSs and their upper and strong level subsets are also provided.

(inf , sup)-Hesitant Fuzzy Subalgebras of BCK/BCI-Algebras

In this paper, we introduce the concept of \((\inf, \sup)\)-hesitant fuzzy subalgebras, which is a general concept of interval-valued fuzzy subalgebras, in BCK/BCI-algebras and investigate its properties. We characterize \((\inf, \sup)\)-hesitant fuzzy subalgebras in terms of sets, fuzzy sets, hesitant fuzzy sets, interval-valued fuzzy sets, Pythagorean fuzzy sets, negative fuzzy sets and bipolar fuzzy sets. Furthermore, characterizations of subalgebras, fuzzy subalgebras, anti-fuzzy subalgebras, negative fuzzy subalgebras, Pythagorean fuzzy subalgebras and bipolar fuzzy subalgebras of BCK/BCI-algebras are given in terms of \((\inf, \sup)\)-hesitant fuzzy subalgebras and interval-valued fuzzy subalgebras.

A New Interpretation of Multi-Polarity Fuzziness Subalgebras of BCK/BCI-Algebras

In this paper, as a further generalization of m-polar fuzziness subalgebras, we present the conception of an m-polar -fuzzy subalgebra in BCK/BCI-algebras and discuss some interesting properties of it. We define an m-polar -fuzzy subalgebras and explore some of its significant attributes. Certain features of m-polar -fuzzy subalgebras are established.

Linear Diophantine Fuzzy Set Theory Applied to BCK/BCI-Algebras

In this paper, we apply the concept of linear Diophantine fuzzy sets in BCK/BCI-algebras. In this respect, the notions of linear Diophantine fuzzy subalgebras and linear Diophantine fuzzy (commutative) ideals are introduced and some vital properties are discussed. Additionally, characterizations of linear Diophantine fuzzy subalgebras and linear Diophantine fuzzy (commutative) ideals are considered. Moreover, the associated results for linear Diophantine fuzzy subalgebras, linear Diophantine fuzzy ideals and linear Diophantine fuzzy commutative ideals are obtained.

Interval-Valued Intuitionistic Fuzzy Subalgebras/Ideals of Hilbert Algebras

In this paper, the concept of interval-valued intuitionistic fuzzy sets to subalgebras and ideals of Hilbert algebras is introduced. The inverse image of interval-valued intuitionistic fuzzy subalgebras and interval-valued intuitionistic fuzzy ideals of Hilbert algebras is studied and some related properties are investigated. Equivalence relations on interval-valued intuitionistic fuzzy ideals are discussed.

Intuitionistic fuzzy Hilbert algebras

The notions of intuitionistic fuzzy subalgebras and intuitionistic fuzzy ideals of Hilbert algebras are introduced and studied in this work, as well as some of their properties. Under intuitionistic fuzzy ideals, we also investigate inverse images of homomorphisms. Finally, several equivalence relations on the class of all intuitionistic fuzzy ideals are examined.

The ( 2 , 3 ) -fuzzy set and its application in BCK-algebras and BCI-algebras

It is well-known that an intuitionistic fuzzy set is a generalization of a fuzzy set. Intuitionistic fuzzy sets deal with two types of fuzzy sets, namely membership function and non-membership function, under the condition that the sum of the membership degree and non-membership degree is less than or equal to \(1\). If the sum of membership degree and non-membership degree is greater than or equal to \(1\), the intuitionistic fuzzy set feels limited in its role. The Pythagorean fuzzy set that emerged to overcome these limitations is the generalization of the intuitionistic fuzzy set. As another form of generalization of the intuitionistic fuzzy set, the concept of the \((2,3)\)-fuzzy set is introduced in this article and several attributes are investigated. The semigroup structure is assigned to the collection of \((2,3)\)-fuzzy sets. The concepts of \((2,3)\)-fuzzy subalgebra for BCK/BCI-algebra and closed \((2,3)\)-fuzzy subalgebra for BCI-algebra are introduced and their properties are investigated. The relationship between the \((2,3)\)-fuzzy subalgebra and the degree function is discussed. A new \((2,3)\)-fuzzy subalgebra is generated using the given \((2,3)\)-fuzzy subalgebra. The union and intersection of \((2,3)\)-fuzzy subalgebras are addressed, and the characterization of the \((2,3)\)-fuzzy subalgebra using the \((2,3)\)-cutty set is addressed. Conditions for closing the \((2,3)\)-fuzzy subalgebra in the BCI-algebra are retrieved.

Fuzzy Set Theoretic Approach to Generalized Ideals in BCK/BCI-Algebras

This paper deals with the study of generalizations of fuzzy subalgebras and fuzzy ideals in BCK/BCI-algebras. In fact, the notions of ∈ , ∈ ∨ κ ~ ∗ , q κ ~ -fuzzy subalgebras, ∈ , ∈ ∨ κ ~ ∗ , q κ ~ -fuzzy ideals, and ∈ ∨ κ ~ ∗ , q κ ~ , ∈ ∨ κ ~ ∗ , q κ ~ -fuzzy ideals in BCK/BCI-algebras are introduced. Some examples are provided to demonstrate the logic of the concepts used in this paper. Moreover, some characterizations of these notions are discussed. In addition, the concept of ∈ , ∈ ∨ κ ~ ∗ , q κ ~ -fuzzy commutative ideals is introduced, and several significant characteristics are discussed. It is shown that for a BCK-algebra A , every ∈ , ∈ ∨ κ ~ ∗ , q κ ~ -commutative ideal of a BCK-algebra is an ∈ , ∈ ∨ κ ~ ∗ , q κ ~ -fuzzy ideal, but the converse does not hold in general; a counter example is constructed.