Search Results (7)

To systematically address the intricate multiple criteria decision-making (MCDM) challenges to practical situations where uncertain and hesitant information plays a critical role in guiding optimal choices. In this article, we introduce the concept of $\mathfrak{m}$-polar Q-hesitant fuzzy ($\mathcal{MPQHF}$) $\mathfrak{B}\mathrm{C}\mathbb{K}/\mathfrak{B}\mathrm{C}\mathbb{I}$ algebras, combining $\mathfrak{m}$-PFS theory with Q-hesitant fuzzy set theory in the framework of $\mathfrak{B}\mathrm{C}\mathbb{K}$/$\mathfrak{B}\mathrm{C}\mathbb{I}$ algebras. This innovative approach enhances the attitudes of uncertainty, vagueness, and hesitance of data in decision-making processes. We investigate the features and actions of this proposed hybrid approach to fuzzy sets and hesitant fuzzy sets, focusing on $\mathcal{MPQHF}$ subalgebras, and explore the characteristics of several kinds of ideals under $\mathfrak{B}\mathrm{C}\mathbb{K}$/$\mathfrak{B}\mathrm{C}\mathbb{I}$ algebras. It also showed that it can better represent complex levels of uncertainty than regular sets. The proposed method’s theoretical framework offers a better way to show uncertain data in areas like engineering, computer science, and computational mathematics. By linking theoretical advancements of $\mathcal{MPQHF}$ sets with practical applications, we highlight the benefits and challenges of this approach. Demonstrating the practical uses of the $\mathcal{MPQHF}$ sets aims to encourage broader adoption. Symmetry plays a vital role in algebraic structure and is used in various fields like decision-making, encryption, pattern recognition problems, and automata theory. Furthermore, this work enhances the understanding of algebraic structures and offers a robust tool for career exploration and development through improved decision-making methodologies. Full article

Cubic multipolar structure with finite degree (briefly, cubic k-polar ($C_{k}P$) structure) is a new hybrid extension of both k-polar fuzzy (${}_{k}PF$) structure and cubic structure in which $C_{k}P$ structure consists of two parts; the first one is an interval-valued k-polar fuzzy ($I{V}_{k}PF$) structure acting as a membership grade extended from the interval $\mathbb{P}[0,1]$ to $\mathbb{P}{[0,1]}^k$ (i.e., from interval-valued of real numbers to the k-tuple interval-valued of real numbers), and the second one is a ${}_{k}PF$ structure acting as a nonmembership grade extended from the interval $[0,1]$ to ${[0,1]}^k$ (i.e., from real numbers to the k-tuple of real numbers). This approach is based on generalized cubic algebraic structures using polarity concepts and therefore the novelty of a $C_{k}P$ algebraic structure lies in its large range comparative to both ${}_{k}PF$ algebraic structure and cubic algebraic structure. The aim of this manuscript is to apply the theory of $C_{k}P$ structure on BCK/BCI-algebras. We originate the concepts of $C_{k}P$ subalgebras and (closed) $C_{k}P$ ideals. Moreover, some illustrative examples and dominant properties of these concepts are studied in detail. Characterizations of a $C_{k}P$ subalgebra/ideal are given, and the correspondence between $C_{k}P$ subalgebras and (closed) $C_{k}P$ ideals are discussed. In this regard, we provide a condition for a $C_{k}P$ subalgebra to be a $C_{k}P$ ideal in a BCK-algebra. In a BCI-algebra, we provide conditions for a $C_{k}P$ subalgebra to be a $C_{k}P$ ideal, and conditions for a $C_{k}P$ subalgebra to be a closed $C_{k}P$ ideal. We prove that, in weakly BCK-algebra, every $C_{k}P$ ideal is a closed $C_{k}P$ ideal. Finally, we establish the $C_{k}P$ extension property for a $C_{k}P$ ideal. Full article

In 2020, Kang et al. introduced the concept of a multipolar intuitionistic fuzzy set of finite degree, which is a generalization of a k-polar fuzzy set, and applied it to a BCK/BCI-algebra. The specific purpose of this study was to apply the concept of a multipolar intuitionistic fuzzy set of finite degree to a hyper BCK-algebra. The notions of the k-polar intuitionistic fuzzy hyper BCK-ideal, the k-polar intuitionistic fuzzy weak hyper BCK-ideal, the k-polar intuitionistic fuzzy s-weak hyper BCK-ideal, the k-polar intuitionistic fuzzy strong hyper BCK-ideal and the k-polar intuitionistic fuzzy reflexive hyper BCK-ideal are introduced herein, and their relations and properties are investigated. These concepts are discussed in connection with the k-polar lower level set and the k-polar upper level set. Full article

In 2020, Kang, Song and Jun introduced the notion of multipolar intuitionistic fuzzy set with finite degree, which is a generalization of intuitionistic fuzzy set, and they applied it to BCK/BCI-algebras. In this paper, we used this notion to study p-ideals of BCI-algebras. The notion of k-polar intuitionistic fuzzy p-ideals in BCI-algebras is introduced, and several properties were investigated. An example to illustrate the k-polar intuitionistic fuzzy p-ideal is given. The relationship between k-polar intuitionistic fuzzy ideal and k-polar intuitionistic fuzzy p-ideal is displayed. A k-polar intuitionistic fuzzy p-ideal is found to be k-polar intuitionistic fuzzy ideal, and an example to show that the converse is not true is provided. The notions of p-ideals and k-polar $(\in ,\in )$ -fuzzy p-ideal in BCI-algebras are used to study the characterization of k-polar intuitionistic p-ideal. The concept of normal k-polar intuitionistic fuzzy p-ideal is introduced, and its characterization is discussed. The process of eliciting normal k-polar intuitionistic fuzzy p-ideal using k-polar intuitionistic fuzzy p-ideal is provided. Full article

When events occur in everyday life, it is sometimes advantageous to approach them in two directions to find a solution for them. As a mathematical tool to handle these things, we can consider the intuitionistic fuzzy set. However, when events are complex and the key to a solution cannot be easily found, we feel the need to approach them for hours and from various directions. As mathematicians, we wish we had the mathematical tools that apply to these processes. If these mathematical tools were developed, we would be able to apply them to algebra, topology, graph theory, etc., from a close point of view, and we would be able to apply these research results to decision-making and/or coding theory, etc., from a distant point of view. In light of this view, the purpose of this study is to introduce the notion of a multipolar intuitionistic fuzzy set with finite degree (briefly, k-polar intuitionistic fuzzy set), and to apply it to algebraic structure, in particular, a BCK/BCI-algebra. The notions of a k-polar intuitionistic fuzzy subalgebra and a (closed) k-polar intuitionistic fuzzy ideal in a BCK/BCI-algebra are introduced, and related properties are investigated. Relations between a k-polar intuitionistic fuzzy subalgebra and a k-polar intuitionistic fuzzy ideal are discussed. Characterizations of a k-polar intuitionistic fuzzy subalgebra/ideal are provided, and conditions for a k-polar intuitionistic fuzzy subalgebra to be a k-polar intuitionistic fuzzy ideal are provided. In a BCI-algebra, relations between a k-polar intuitionistic fuzzy ideal and a closed k-polar intuitionistic fuzzy ideal are discussed. A characterization of a closed k-polar intuitionistic fuzzy ideal is considered, and conditions for a k-polar intuitionistic fuzzy ideal to be closed are provided. Full article

Recent trends in modern information processing have focused on polarizing information, and and bipolar fuzzy sets can be useful. Bipolar fuzzy sets are one of the important tools that can be used to distinguish between positive information and negative information. Positive information, for example, already observed or experienced, indicates what is guaranteed to be possible, and negative information indicates that it is impossible, prohibited, or certainly false. The purpose of this paper is to apply the bipolar fuzzy set to BCK/BCI-algebras. The notion of (translated) k-fold bipolar fuzzy sets is introduced, and its application in BCK/BCI-algebras is discussed. The concepts of k-fold bipolar fuzzy subalgebra and k-fold bipolar fuzzy ideal are introduced, and related properties are investigated. Characterizations of k-fold bipolar fuzzy subalgebra/ideal are considered, and relations between k-fold bipolar fuzzy subalgebra and k-fold bipolar fuzzy ideal are displayed. Extension of k-fold bipolar fuzzy subalgebra is discussed. Full article

Multi-polar vagueness in data plays a prominent role in several areas of the sciences. In recent years, the thought of m-polar fuzzy sets has captured the attention of numerous analysts, and research in this area has escalated in the past four years. Hybrid models of fuzzy sets have already been applied to many algebraic structures, such as $BCK/BCI$ -algebras, lie algebras, groups, and symmetric groups. A symmetry of the algebraic structure, mathematically an automorphism, is a mapping of the algebraic structure onto itself that preserves the structure. This paper focuses on combining the concepts of m-polar fuzzy sets and m-polar fuzzy points to introduce a new notion called m-polar $(\alpha ,\beta )$ -fuzzy ideals in $BCK/BCI$ -algebras. The defined notion is a generalization of fuzzy ideals, bipolar fuzzy ideals, $(\alpha ,\beta )$ -fuzzy ideals, and bipolar $(\alpha ,\beta )$ -fuzzy ideals in $BCK/BCI$ -algebras. We describe the characterization of m-polar $(\in ,\in \vee q)$ -fuzzy ideals in $BCK/BCI$ -algebras by level cut subsets. Moreover, we define m-polar $(\in ,\in \vee q)$ -fuzzy commutative ideals and explore some pertinent properties. Full article