A useful tool for expressing fuzziness and uncertainty is the recently developed n,m-rung orthopair fuzzy set (n,m-ROFS). Due to their superior ability to manage uncertain situations compared to theories of q-rung orthopair fuzzy sets, the n,m-rung orthopair fuzzy sets have variety of applications in decision-making in daily life. To deal with ambiguity and unreliability in multi-attribute group decision-making, this study introduces a novel tool called $$k^{n}_{m}$$k m n -rung picture fuzzy set ( $$k^{n}_{m}$$k m n -RPFS). The suggested $$k^{n}_{m}$$k m n -RPFS incorporates all of the benefits of n,m-ROFS and represents both the quantitative and qualitative analyses of the decision-makers. The presented model is a fruitful advancement of the q-rung picture fuzzy set (q-RPFS). Furthermore, numerous of its key notions, including as complement, intersection, and union are explained and illustrated with instances. In many decision-making situations, the main benefit of $$k^{n}_{m}$$k m n -rung picture fuzzy sets is the ability to express more uncertainty than q-rung picture fuzzy sets. Then, along with their numerous features, we discover the basic set of operations for the $$k^{n}_{m}$$k m n -rung picture fuzzy sets. Importantly, we present a novel operator, $$k^{n}_{m}$$k m n -rung picture fuzzy weighted power average ( $$k^{n}_{m}$$k m n -RPFWPA) over $$k^{n}_{m}$$k m n -rung picture fuzzy sets, and use it to multi-attribute decision-making issues for evaluating alternatives with $$k^{n}_{m}$$k m n -rung picture fuzzy information. Additionally, we use this operator to pinpoint the countries with the highest expat living standards and demonstrate how to choose the best option by comparing aggregate findings and applying score values. Finally, we compare the outcomes of the q-RPFEWA, SFWG, PFDWA, SFDWA, and SFWA operators to those of the $$k^{n}_{m}$$k m n -RPFWPA operator.

Fuzzy set (FS) theory and rough sets (RSs) are constructed to accommodate the data uncertainty. In contrast, the bipolar FS (BFS) theory can tackle the uncertainty and the bipolarity of the data in different circumstances. This article aims to introduce the idea of rough bipolar fuzzy ideals in semigroup (SG), which is a generalization of the concept of rough BFSs (RBFSs) in an SG. We also investigate the roughness in the bipolar fuzzy subsemigroup (BF-SSG) with the help of congruence relation (cng-R) defined on the SG and studied some relevant structural properties. Moreover, the idea is extended to the rough bipolar fuzzy left ideal, rough bipolar fuzzy right ideal, rough bipolar fuzzy two-sided ideal, rough bipolar fuzzy interior ideal, and rough bipolar fuzzy bi-ideal in SGs. Further, it is seen that cng-Rs and complete cng-Rs play vital roles in the construction of rough approximations of bipolar fuzzy ideals. Consequently, their associated properties are explored by using cng-Rs and complete cng-Rs.

In this paper, we hybridize the bipolar fuzzy set (BFS) theory with T-rough sets (T-RSs) and initiate the novel idea of T-rough BFSs (T-RBFSs). The concept presented in this article has never been discussed earlier. Moreover, we investigate the axiomatic systems of T-RBFSs in detail. Meanwhile, we address a decision-making (DM) problem having data endowed with fuzziness and bipolarity in the framework of the T-RBFSs. We also propose an algorithm for this application. This algorithm facilitates tackling the case when there is a team of decision-makers instead of a single decision-maker and when the objects of one set need to be approximated by grading the objects of some other set. Moreover, a practical application of T-RBFSs in DM problems is given, accompanied by a practical example, which provides the optimal and the worst decision between some objects. Finally, a comparative analysis of the recommended study with several prevailing approaches is given to endorse the advantages of the suggested research.

The aim of this study is to introduce and investigate two new classes of separation axioms called supra soft \(R_0\) and supra soft \(R_1\). They are defined in the spaces of supra soft topologies by using the notions of supra soft open sets and supra soft closure operator. We discuss the basic properties and characterizations of them. We also study the relationships between these classes and some other supra soft separation axioms with many results and explanative examples. Moreover, the connections between the properties of these classes and those in some generated soft topologies are presented. Finally, we show that these classes are preserved under subspaces, which means they are supra soft topological properties.

One of the known approaches to studying topological concepts is to utilize subclasses of topology, such as clopen sets and generalized closed sets. In this study, we apply the notion of fuzzy generalized $\mu$-closed sets ($Fg\mu$-closed sets) to establish and analyze novel categories of spaces, namely $Fg\mu $-regular, $Fg\mu $-normal, and $F\mu $-symmetric spaces in the frame of generalized fuzzy topology($GFT$). We investigate the fundamental properties of these classes, exploring their unique characteristics and preservation theorems under $Fg\mu$-continuous maps. We establish the interrelationships between these classes and the other separation axioms in this setting, and we demonstrated that $F\mu $-regular, $F\mu $-normal, and $F\mu $-symmetric spaces are special cases of $Fg\mu $-regular, $Fg\mu $-normal, and $F\mu$-$T_{1}$ spaces respectively. Additionally, we show that the equivalence for these cases hold when the $GFT$ is $F\mu$-$T_{\frac{1}{2}}$. The connections between these classes and their counterparts in the crisp $GT$ are studied. Finally, we discuss these classes' hereditary and topological properties, further enhancing our comprehension of their behavior and implications.

In this article, we study the concept of finite open sets in the spaces of supra soft-topologies. We aim by defining this concept to furnish the researchers with a tool they generate new forms of topological concepts via supra soft-topologies such as continuity, compactness, separation axioms, etc. We discuss the main properties of this concept and prove that this class of soft sets is neither infra soft-topology nor supra soft-topology. Also, we reveal its connections with other celebrated generalizations of supra soft-open sets and conclude the conditions they guarantee the equivalence between them. After that, we display some soft operators, like interior and closure, inspired by the classes of supra finite soft-open and supra finite soft-closed sets. Finally, we set forth some types of continuity defined by these classes and describe their main characterizations and behaviors in different cases such as decomposition theorems and transition between the realms of of supra soft-topologies and their parametric supra topologies. The obtained results and implementations are articulated with the aid of some counterexamples.