Soft Ideals Research Articles

Fuzzy Soft Ideals of Fuzzy Soft Ternary Γ- Semirings

Fuzzy Soft Ideals of Fuzzy Soft Ternary Γ- Semirings

Cubic Soft Ideals on B-algebra for Solving Complex Problems: Trend Analysis, Proofs, Improvements, and Applications

Cubic Soft Ideals on B-algebra for Solving Complex Problems: Trend Analysis, Proofs, Improvements, and Applications

A Study on Rough Fuzzy Bipolar Soft Ideals in Semigroups

The theories of rough sets (RSs) and soft sets (SSs) are practical mathematical techniques to accommodate data uncertainty. On the other hand, fuzzy bipolar soft sets (FBSSs) can address uncertainty and bipolarity in various situations. The key objective of this study is to establish the notions of rough fuzzy bipolar soft ideals in semigroup (SG), which is an extension of the idea of rough fuzzy bipolar soft sets in a SG. Also, we have analyzed the roughness in the bipolar fuzzy subsemigroup (BF-SSG) by employing a congruence relation (CR) defined on the SG and investigating several related characteristics. Further, the idea is expanded to the rough fuzzy bipolar soft ideal, rough fuzzy bipolar soft interior ideal, and rough fuzzy bipolar soft bi-ideal in SGs. Moreover, it is observed that CR and complete CR (CCR) are critical in developing rough approximations of fuzzy bipolar soft ideals. Therefore, their related characteristics are studied via CRs and CCRs.

Connectedness of Soft-Ideal Topological Spaces

Despite its apparent simplicity, the idea of connectedness has significant effects on topology and its applications. An essential part of the intermediate-value theorem is the idea of connectedness. In many applications, such as population modeling, robotics motion planning, and geographic information systems, connectedness is significant, and it is a critical factor in differentiating between various topological spaces. This study uses soft open sets and the concept of soft ideals as a new class of soft sets to present and explore the ideas of soft connected spaces and strongly soft connected spaces with soft ideals. Also, under certain assumptions regarding the subsequent concepts—soft-ideal connectedness and stronglysoft-ideal connectedness in soft-ideal topological spaces—we characterize this new class of sets by employing soft open sets and soft ideals to examine its fundamental features. Furthermore, we look at a symmetry between our new notions and other existing ones, and this study examines the relationships between these concepts.

Soft closure spaces via soft ideals

<abstract><p>This paper was devoted to defining new soft closure operators via soft relations and soft ideals, and consequently new soft topologies. The resulting space is a soft ideal approximation. Many of the well known topological concepts were given in the soft set-topology. Particularly, it introduced the notations of soft accumulation points, soft continuous functions, soft separation axioms, and soft connectedness. Counterexamples were introduced to interpret the right implications. Also, a practical application of the new soft approximations was explained by an example of a real-life problem.</p></abstract>

On Intuitionistic Fuzzy Neutrosophic Soft Ideal Topological Spaces

The purpose of this paper is to introduce the notion of intuitionistic fuzzy neutronsophic soft ideal in Intuitionistic fuzzy neutronsophic soft set theory. The concept of intuitionistic fuzzy neutrosophic soft local function is also introduced. These concepts are discussed with a view to find new intuitionistic fuzzy neutronsophic soft topologies from the original one. The basic structure, respecially a basic for such generated Intuitionistic fuzzy neutronsophic soft topologies also studied here. Finally, the notion of compatibility of intuitionistic fuzzy neutronsophic soft ideals with Intuitionistic fuzzy neutrosophic soft topologies is introduced and some equivalent conditions concerning, this topic are established here.

N-polar Z-hesitant Complementary Fuzzy Soft Set in BCK/BCI-Algebras

This paper introduces an innovative concept known as n-polar Z-hesitant Anti-Fuzzy Soft Sets (MZHAFSs) within the framework of BCK/BCI-algebras. Soft set theory originates in the captivating field of fuzzy set theory. Our approach is a harmonious synthesis of n-polar anti-fuzzy set theory, soft set models, and Z-hesitant anti-fuzzy sets, skillfully applied within the framework of BCK/BCI-algebras. This effort leads to the introduction of a new variant of fuzzy sets termed MZHAFSs (n-polar Z-hesitant anti-fuzzy soft sets) in the context of BCK/BCI-algebras. Additionally, we elucidate the concept of n-polar Z-hesitant anti-fuzzy soft sets to provide a comprehensive understanding. Furthermore, we introduce and define various related concepts, including n-polar Z-hesitant anti-fuzzy soft subalgebras, n-polar Z-hesitant anti-fuzzy soft ideals, n-polar Z-hesitant anti-fuzzy soft closed ideals, and n-polar Z-hesitant anti-fuzzy soft commutative ideals, and establish meaningful connections between them. We also present and rigorously prove several theorems that are pertinent to these newly introduced notions.

An Abstract Context to Lattice-Based Ideals (Filters) with Multi-Fuzzy Soft Settings

In this study, multi-fuzzy soft ideals (filters) are introduced over the lattice-ordered multi-fuzzy soft set ([Formula: see text]) and their results are shown. The ideas of lattice-ordered multi-fuzzy soft homomorphism ([Formula: see text]) between two [Formula: see text]s are then introduced, along with those of lattice-ordered multi-fuzzy soft isotone map [Formula: see text], lattice-ordered multi-fuzzy soft isomorphism ([Formula: see text]), and kernel of [Formula: see text]. We also look at the principal ideal and soft prime ideal of [Formula: see text].

Congruence Representations via Soft Ideals in Soft Topological Spaces

This article starts with a study of the congruence of soft sets modulo soft ideals. Different types of soft ideals in soft topological spaces are used to introduce new weak classes of soft open sets. Namely, soft open sets modulo soft nowhere dense sets and soft open sets modulo soft sets of the first category. The basic properties and representations of these classes are established. The class of soft open sets modulo the soft nowhere dense sets forms a soft algebra. Elements in this soft algebra are primarily the soft sets whose soft boundaries are soft nowhere dense sets. The class of soft open sets modulo soft sets of the first category, known as soft sets of the Baire property, is a soft σ-algebra. In this work, we mainly focus on the soft σ-algebra of soft sets with the Baire property. We show that soft sets with the Baire property can be represented in terms of various natural classes of soft sets in soft topological spaces. In addition, we see that the soft σ-algebra of soft sets with the Baire property includes the soft Borel σ-algebra. We further show that soft sets with the Baire property in a certain soft topology are equal to soft Borel sets in the cluster soft topology formed by the original one.

Bipolar Intuitionistic Fuzzy Soft Ideals of BCK/BCI-Algebras and Its Applications in Decision-Making

In this paper, we merge the concepts of soft set theory and a bipolar intuitionistic fuzzy set. A bipolar intuitionistic fuzzy soft ideal in a BCK-algebra is defined as a soft set over the set of elements in the BCK-algebra, with each element associated with an intuitionistic fuzzy set. This relationship captures degrees of uncertainty, hesitancy, and non-membership degrees within the context of BCK-algebras. We investigate basic operations on bipolar intuitionistic fuzzy soft ideals such as union, intersection, AND, and OR. The intersection, union, AND, and OR of two bipolar intuitionistic fuzzy soft ideals is a bipolar intuitionistic fuzzy soft ideal. We also demonstrate how to use a bipolar intuitionistic fuzzy soft set to solve a problem involving decision making. Finally, we provide a general approach for handling decision-making problems using a bipolar intuitionistic fuzzy soft set.

Fuzzy Soft Boolean Rings

This article introduces the idea of (fuzzy soft Boolean rings) FSBRs and investigates their algebraic properties. The concepts of fuzzy soft ideals (FSIs) of FSBRs, and idealistic fuzzy soft Boolean rings (IFSBRs) are then defined and discussed.

A Novel Characterization of Fuzzy Soft Substructures in Quantales Theory

In this paper, we use an algebraic structure quantale and define the idea of fuzzy soft substructures as a generalization of fuzzy substructures in quantale. These fuzzy soft substructures include fuzzy soft subquantales, fuzzy soft ideals, fuzzy soft prime ideals, fuzzy soft semiprime ideals, and fuzzy soft primary ideals. Furthermore, different characterizations of fuzzy soft substructures in quantales are introduced. Moreover, we extend this ideology to investigate that for each fuzzy soft substructure in quantale, there exists an α-soft substructure in quantales. These fuzzy soft subquantales and fuzzy soft ideals are characterized by their level subquantales and ideals, respectively. Finally, fuzzy soft image and fuzzy soft inverse image of fuzzy soft substructures under quantale homomorphism in quantale are discussed.

Soft Sub-nearsemirings, Soft Ideals and Soft S-Subsemigroups of Nearsemirings

Soft Sub-nearsemirings, Soft Ideals and Soft S-Subsemigroups of Nearsemirings

Hybrid Nil Radical of a Ring

The nature of universe problems is ambiguous due to the presence of asymmetric data in almost all disciplines, including engineering, mathematics, medical sciences, physics, computer science, operations research, artificial intelligence, and management sciences, and they involve various types of uncertainties when dealing with them on various occasions. To deal with the challenges of uncertainty and asymmetric information, different theories have been developed, including probability, fuzzy sets, rough sets, soft ideals, etc. The strategies of hybrid ideals, hybrid nil radicals, hybrid semiprime ideals, and hybrid products of rings are introduced in this paper and hybrid structures are used to examine the structural properties of rings.

Notes on intuitionistic fuzzy soft ideals in BCK/BCI-algebras

In this paper, we show that the inappropriateness of Example 3.10 and the error of Theorem 4.7 in [M.Balamurugan, G.Balasubramanian, C.Ragavan. Intuitionistic Fuzzy Soft Ideals in BCK/BCI-algebras. Materials today: proceedings 16(2019)496-503.], then give a more perfect example and a counterexample respectively.

Hybrid Ideals in an AG-Groupoid

In general, models of universe problems in almost all fields, such as engineering, mathematics, medical sciences, computer science, physics, management sciences, artificial intelligence, and operations research, are practically full of complexities and include various types of uncertainties when dealing with them on numerous occasions. Different theories, such as probability, rough sets, fuzzy sets, soft ideals, and so on, have been created to deal with these uncertainties. An algebraic structure, AG-groupoid, is an intermediate structure between two types of structures: commutative semigroup and groupoid. This structure has a very close relationship to a commutative semigroup since a commutative AG-groupoid is always a semigroup. These structures have so many applications in flocks theory, geometry, topology, and many more. We explore several structural properties of an AG-groupoid by using hybrid structures in this paper. The main motivation behind this paper is to present the concepts of hybrid ideals, hybrid bi-ideals and hybrid interior ideals of an AG-groupoid and characterize AG-groupoid in terms of hybrid structures. Also, we show that the hybrid intersection and hybrid product structures will coincide under certain conditions.

New approach to DeMorgan’s laws for soft sets via soft ideals

New approach to DeMorgan’s laws for soft sets via soft ideals

More on Prime, Maximal and Principal Soft Ideals of Soft Rings

In this paper, we aim to extend the studies [M. R. Alimoradi, R. Rezaei and M. Rahimi, Some notes on ideals in soft rings, Journal Australian Journal of Basic and Applied Sciences 6(3) (2012) 717–721; F. Koyuncu and B. Tanay, Some soft algebraic structures, Journal of New Results Science 10(2016) 38–51] as regards maximal, prime and principal soft ideal of soft rings, characterize soft rings with these soft ideals and also provide some more relations between maximal, prime and principal soft ideals of soft rings. The notions of maximality and primeness points of soft ideals of a soft rings are defined, maximal and prime idealistic soft rings as well as maximal, prime and principal soft ideal of soft rings and their basic properties are more investigated under certain conditions, especially by means of homomorphism and epimorphism of rings. We apply some of the basic results about maximal ideals and prime ideals in classical abstract algebra to maximal, prime and principal idealistic soft rings and we give some of their interrelations between each others.

Generalization of $$(\in , \in \vee q_{k})$$-fuzzy soft near-rings and $$(\in , \in \vee q_{k})$$-fuzzy soft ideals over near-rings

Generalization of $$(\in , \in \vee q_{k})$$-fuzzy soft near-rings and $$(\in , \in \vee q_{k})$$-fuzzy soft ideals over near-rings

Fuzzy soft set theory applied to commutative ideals of BCK-algebras

In the present paper, we apply the fuzzy soft set theory to commutative ideals of BCK-algebras. In fact, the notion of fuzzy soft commutative ideals over BCK-algebras is introduced, and related properties are investigated. Relations between fuzzy soft ideals and fuzzy soft commutative ideals are discussed, and conditions for a fuzzy soft ideal to be a fuzzy soft commutative ideal are provided. The “AND” operation, “extended intersection” and “union” of fuzzy soft (commutative) ideals are studied. Furthermore, characterizations of fuzzy soft (commutative) ideals are considered.