This paper presents N-bipolar soft expert (N-BSE) sets, a novel framework designed to enhance multi-attribute group decision-making (MAGDM) by incorporating expert input, bipolarity, and non-binary evaluations. Existing MAGDM approaches often lack the ability to simultaneously integrate positive and negative assessments, especially in nuanced, multi-valued evaluation spaces. The proposed N-BSE model addresses this limitation by offering a comprehensive, mathematically rigorous structure for decision-making (DM). Fundamental operations of the N-BSE model are defined and analyzed, ensuring its theoretical consistency and applicability. To demonstrate its practical utility, the N-BSE model is applied to a general case study on sustainable energy solutions, illustrating its effectiveness in handling complex DM scenarios. An algorithm is proposed to streamline the DM process, enabling systematic and transparent identification of optimal alternatives. Additionally, a comparative analysis emphasizes the advantages of the N-BSE model over existing MAGDM frameworks, highlighting its capacity to integrate diverse expert opinions, evaluate both positive and negative attributes, and support multi-valued assessments. By bridging the gap between theoretical development and practical application, this paper contributes to advancing DM methodologies.

Group theory is the part of mathematics which addresses the study of symmetry. This paper extends the investigation of the soft group theory which Aktaş and Çağman have defined. We study new concepts in the soft group theory such as the center of the soft group, the kernel of soft homomorphism and soft automorphisms with their basic properties. Furthermore, the concept of soft point groups is introduced and the properties of these soft groups are studied. We state the concept of characteristic soft subgroups of a given soft group. Also, some theorems related to this concept are investigated. We study the characteristic soft subgroups of a given soft point group. The characteristic soft subgroups play a significant part in the study of the soft group theory and are useful for understanding the structure of a soft group and its soft automorphisms. As an application of characteristic soft subgroups, they allow us to identify and study important soft subgroups that are preserved under soft automorphisms. Also, practical applications for our theory can be conducted in future work such as the relation with other disciplines in sciences.

Multiple criteria decision-making (MCDM) is a significant area of decision-making theory that certainly warrants attention. It might be difficult to accurately convey the necessary decision facts when navigating decision-making problems since we frequently run into complicated issues and unpredictable situations. To address this, introducing the novel idea of the bipolar pqr-spherical fuzzy set (${\mathrm{B}}_{\mathrm{pqr}}$SFS), a hybrid structure of the bipolar fuzzy set (BFS) and the pqr-spherical fuzzy set (pqr-SFS), is the main goal of this work. The fundamental (set-theoretic and algebraic) operations on ${\mathrm{B}}_{\mathrm{pqr}}$SFSs are explained as well as their relations to several known models. A distance measure, such as Euclidean distance, among ${\mathrm{B}}_{\mathrm{pqr}}$SFNs, is provided. Afterward, we expand the fundamental aggregation operators to the pqr-spherical fuzzy (${\mathrm{B}}_{\mathrm{pqr}}$SF) environment by developing bipolar pqr-spherical fuzzy-weighted averaging and bipolar pqr-spherical fuzzy-weighted geometric operators for aggregating ${\mathrm{B}}_{\mathrm{pqr}}$SFNs. According to the aforementioned distance measure and operators, an MCDM approach is established consisting of two algorithms, namely, the TOPSIS method and the method using the proposed operators in the ${\mathrm{B}}_{\mathrm{pqr}}$SF context. Moreover, a numerical example is provided in order to ensure that the presented model is applicable. By using the two algorithms, a comparative analysis of the proposed method with other existing ones is given in order to verify the feasibility of the suggested decision-making procedure.

To address the complexity of daily problems, soft set theory has emerged as a valuable tool, providing innovative mathematical techniques to manage vast amounts of data and ambiguity. The study of soft topology involves the investigation of various properties of soft sets and functions, as well as the development of new mathematical models and techniques for addressing uncertainty. The main motivation of this paper is to delve deeper into the subject and devise new methodologies to address real-world challenges more effectively and unlock the full potential of soft sets in various applications. In this paper, we present a novel soft topology, which is constructed using soft single points on a nonempty set V in relation to a topology on $V.$ We investigate and study the behaviors and properties associated with this particular type of soft topology. Furthermore, we shed light on the soft separation axioms with this type of soft topology and investigate whether these axioms are inherited from the corresponding ordinary topology or not. Our study is concerned with examining the connection between ordinary topologies and the soft topologies generated that arise from them, with the aim of identifying their interdependencies and potential implications. By studying the connection between soft topologies and their corresponding ordinary topologies, researchers are able to gain a deeper understanding of the properties and behaviors of these structures and develop new modeling approaches for dealing with uncertainty and complexity in data.

Supplier evaluation in supply chain management involves uncertainty, conflicting expert opinions, and criteria with both positive and negative aspects. Traditional fuzzy models cannot simultaneously handle interval uncertainty, bipolar reasoning, and fractional control. To address this limitation, we propose the interval-valued bipolar q-fractional fuzzy set model, which integrates bipolar fuzzy sets (to capture positive and negative membership degrees), interval-valued fuzzy sets (to handle expert imprecision), and q-fractional fuzzy sets (to flexibly control the balance between membership and non-membership degrees). We define the fundamental operations of the model and examine their algebraic properties. Two aggregation operators–the interval-valued bipolar q-fractional fuzzy weighted averaging operator and the interval-valued bipolar q-fractional fuzzy weighted geometric operator–are developed to aggregate such information while preserving idempotency, monotonicity, and boundedness. A systematic ranking procedure applied to a supplier evaluation problem demonstrates the model’s effectiveness in handling conflicting, interval-based assessments. Comparative and sensitivity analyses further reveal that the proposed model provides stable rankings and reliably identifies the best alternative across different parameter settings, confirming its robustness and practical applicability in multi-criteria decision-making for supply chain management.

In a soft environment, various structures associated with sets and topological spaces have been rigorously analyzed, including soft open (closed) sets, soft separation axioms, soft connectedness, and so on. The practical applications of soft set theory underscore its significant impact on real-world problems, offering innovative solutions that enhance decision-making processes across diverse domains. In this study, we built upon existing research by introducing new concepts within the realm of soft spaces, which served as an extension of classical topology. Our focus was on soft isolated sets and soft dense sets (spaces) in themselves. We investigated the fundamental properties and characterizations of these concepts, bolstered by rigorous proofs of theorems and relevant counterexamples. We explored the interrelations between these concepts and the notion of soft closure. Additionally, we showed that a soft topology was classified as soft dense in itself if, and only if, each of its parametric topologies was similarly classified. We also examined the behavior of soft dense spaces in themselves under various operations, including the formation of soft subspaces, the soft topological sum, and the image and inverse image under specific soft mappings.