Many models of uncertain knowledge have been designed that combine expanded views of fuzziness (expressions of partial memberships) with parameterization (multiple subsethood indexed by a parameter set). The standard orthopair fuzzy soft set is a very general example of this successful blend initiated by fuzzy soft sets. It is a mapping from a set of parameters to the family of all orthopair fuzzy sets (which allow for a very general view of acceptable membership and non-membership evaluations). To expand the scope of application of fuzzy soft set theory, the restriction of orthopair fuzzy sets that membership and non-membership must be calibrated with the same power should be removed. To this purpose we introduce the concept of $(a,b)$-fuzzy soft set, shortened as $(a,b)$-FSS. They enable us to address situations that impose evaluations with different importances for membership and non-membership degrees, a problem that cannot be modeled by the existing generalizations of intuitionistic fuzzy soft sets. We establish the fundamental set of arithmetic operations for $(a,b)$-FSSs and explore their main characteristics. Then we define aggregation operators for $(a,b)$-FSSs and discuss their main properties and the relationships between them. Finally, with the help of suitably defined scores and accuracies we design a multi-criteria decision-making strategy that operates in this novel framework. We also analyze a decision-making problem to endorse the validity of $(a,b)$-FSSs for decision-making purposes.

Data uncertainty is a barrier in the decision-making (DM) process. The rough set (RS) theory is an effective approach to study the uncertainty in data, while bipolar soft sets (BSSs) can handle the vagueness and uncertainty as well as the bipolarity of the data in a variety of situations. In this article, we introduce the idea of rough bipolar soft sets (RBSSs) and apply them to find the best decision in two different DM problems in medical science. The first problem is about deciding between the risk factors of a disease. Our algorithm facilitates the doctors to investigate which risk factor is becoming the most prominent reason for the increased rate of disease in an area. The second problem is deciding between the different compositions of a medicine for a particular illness having different effects and side effects. We also propose algorithms for both problems.

The rough set (RS) and multi-granulation RS (MGRS) theories have been successfully extended to accommodate preference analysis by substituting the equivalence relation (ER) with the dominance relation (DR). On the other hand, the bipolar fuzzy sets (BFSs) are effective tools for handling bipolarity and fuzziness of the data. In this study, with the description of the background of risk decision-making problems in reality, we present $(\alpha ,\beta )$-optimistic multi-granulation bipolar fuzzified preference rough sets ( $(\alpha ,\beta {)}^o$-MG-BFPRSs) and $(\alpha ,\beta )$-pessimistic multi-granulation bipolar fuzzified preference rough sets ( $(\alpha ,\beta {)}^p$-MG-BFPRSs) using bipolar fuzzy preference relation (BFPR). Subsequently, the relevant properties and results of both $(\alpha ,\beta {)}^o$-MG-BFPRSs and $(\alpha ,\beta {)}^p$-MG-BFPRSs are investigated in detail. At the same time, a relationship among the $(\alpha ,\beta )$-BFPRSs, $(\alpha ,\beta {)}^o$-MG-BFPRSs and $(\alpha ,\beta {)}^p$-MG-BFPRSs is given.

The paper presents a novel analysis of interrelations between ordinary (crisp) $\sigma$-algebras and soft $\sigma$-algebras. It is known that each soft $\sigma$-algebra produces a system of crisp (parameterized) $\sigma$-algebras. The other way round is also possible. That is to say, one can generate a soft $\sigma$-algebra from a system of crisp $\sigma$-algebras. Different methods of producing soft $\sigma$-algebras are discussed by implementing two formulas. It is demonstrated how these formulas can be used in practice with the aid of some examples. Furthermore, we study the fundamental properties of soft $\sigma$-algebras. Lastly, we show that elements of a soft $\sigma$-algebra contain information about a specific event.

In this work, we present some concepts that are considered unique ideas for topological structures generated by soft settings. We first define the concept of weakly soft $\alpha$-open subsets and characterize it. It is demonstrated the relationships between this class of soft subsets and some generalizations of soft open sets with the help of some illustrative examples. Some interesting results and relationships are obtained under some stipulations like extended and hyperconnected soft topologies. Then, we introduce the interior and closure operators inspired by the classes of weakly soft $\alpha$-open and weakly soft $\alpha$-closed subsets. We establish their master features and derive some formulas that describe the relations among them. Finally, we study soft continuity with respect to this class of soft subsets and investigate its essential properties. In general, we discuss the systematic relations and results that are missing through the frame of our study. The line adopted in this study will create new roads in the branch of soft topology.

Several tools have been put forth to handle the problem of uncertain knowledge. Pawlak (1982) initiated the concept of rough set theory, which is a completely new tool for solving imprecision and vagueness (uncertainty). The main notions in this theory are the upper and lower approximations. One of the most important aims of this theory is to reduce the vagueness of a concept to uncertainty areas at their borders by decreasing the upper approximations and increasing the lower approximations. So, the object of this study is to propose four types of approximation spaces in rough set theory utilizing ideals and a new type of neighborhoods called "the intersection of maximal right and left neighborhoods". We investigate the master properties of the proposed approximation spaces and demonstrate that these spaces reduce boundary regions and improve accuracy measures. A comparative study of the present methods and the previous ones is given and shown that the current study is more general and accurate. The importance of the current paper is not only that it is introducing new kinds of approximation spaces relying mainly on ideals and a new type of neighborhoods which increases the accuracy measure and reduces the boundary region of subsets, but also that these approximation spaces are monotonic, which means that it can be successfully used to evaluate the uncertainty in the data. In the end of this paper, we provide a medical example of the heart attacks problem to show the efficiency of the current techniques in terms of approximation operators, accuracy measures, and monotonic property.

Rough set theory is an advanced uncertainty tool that is capable of processing sophisticated real-world data satisfactorily. Rough approximation operators are used to determine the confirmed and possible data that can be obtained by using subsets. Numerous rough approximation models, inspired by neighborhood systems, have been proposed in earlier studies for satisfying axioms of Pawlak approximation spaces (P-approximation spaces) and improving the accuracy measures. This work provides a formulation a novel type of generalized approximation spaces (G-approximation spaces) based on new neighborhood systems inspired by ${\mathbb{I}}_j$-neighborhoods and ideal structures. The originated G-approximation spaces are offered to fulfill the axiomatic requirements of P-approximation spaces and give more information based on the data subsets under study. That is, they are real simulations of the P-approximation spaces and provide more accurate decisions than the previous models. Several examples are provided to compare the suggested G-approximation spaces with existing ones. To illustrate the application potentiality and efficiency of the provided approach, a numerical example for Chikungunya disease is presented. Ultimately, we conclude our study with a summary and direction for further research.

In this paper, we present a novel family of soft sets named "finite soft-open sets". The purpose of investigating this kind of soft sets is to offer a new tool to structure topological concepts that are stronger than their existing counterparts produced by soft-open sets and their well-known extensions, as well as to provide an environment that preserves some topological characteristics that have been lost in the structures generated by celebrated extensions of soft-open sets, such as the distributive property of a soft union and intersection for soft closure and interior operators, respectively. We delve into a study of the properties of this family and explore its connections with other known generalizations of soft-open sets. We demonstrate that this family strictly lies between the families of soft-clopen and soft-open sets and derive under which conditions they are equivalent. One of the unique features of this family that we introduce is that it constitutes an infra soft topology and fails to be a supra soft topology. Then, we make use of this family to exhibit some operators in soft settings, i.e., soft $fo$-interior, $fo$-closure, $fo$-boundary, and $fo$-derived. In addition, we formulate three types of soft continuity and look at their main properties and how they behave under decomposition theorems. Transition of these types between realms of soft topologies and classical topologies is examined with the help of counterexamples. On this point, we bring to light the role of extended soft topologies to validate the properties of soft topologies by exploring them for classical topologies and vice-versa.

In the present paper, we introduce and discuss a new set of separation properties in fuzzy soft topological spaces called $FS\delta$-separation and $FS\delta$-regularity axioms by using fuzzy soft $\delta$-open sets and the quasi-coincident relation. We provide a comprehensive study of their properties with some supporting examples. Our analysis includes more characterizations, results, and theorems related to these notions, which contributes to a deeper understanding of fuzzy soft separability properties. We show that the $FS\delta$-separation and $FS\delta$-regularity axioms are harmonic and heredity property. Additionally, we examine the connections between $FS{\delta }^{\ast }$-compactness and $FS\delta$-separation axioms and explore the relationships between them. Overall, this work offers a new perspective on the theory of separation properties in fuzzy soft topological spaces, as well as provides a robust foundation for further research in the transmission of properties from fuzzy soft topologies to fuzzy and soft topologies and vice-versa by swapping between the membership function and characteristic function in the case of fuzzy topology and the set of parameters and a singleton set in the case of soft topology.

One popular approach to studying topological concepts is to employ a subclass of topology, such as clopen, regular open, and $\delta$-open sets. Motivated by the advantages of soft topology over classical topologies, we investigate some of these classes in a soft setting. We begin this manuscript by exploring further properties of soft regular open and soft $\delta$-open sets in the context of soft subspaces and soft mappings, as well as describing their behavior in relation to classical topologies. We demonstrate that the condition of an extended soft topology guarantees the symmetry between $\delta$-open sets and soft $\delta$-open sets the realms of soft topology and its crisp topologies. Then, we apply the concept of soft $\delta$-open sets to establish two new classes of soft compactness, namely soft $\delta$-compactness, and soft local $\delta$-compactness. We research the basic properties of these classes, including characterizations and preservation theorems under soft $\delta$-continuous mappings. We reveal the relationship between soft compactness, soft $\delta$-compactness and soft local $\delta$-compactness, and also prove the equivalence between these concepts when the soft topology is soft regular. Finally, the symmetry between our new classes and their counterparts in some classical topologies is studied amply; especially, when the soft topology is extended or stable. The implementations of the current results and relationships are elucidated by some supporting examples.