The current work concentrates on generating different topologies by using the concept of the ideal. These topologies are used to make more thorough studies on generalized rough set theory. The rough set theory was first proposed by Pawlak in 1982. Its core concept is upper and lower approximations. The principal goal of the rough set theory is reducing the vagueness of a concept to uncertainty areas at their borders by increasing the lower approximation and decreasing the upper approximation. For the mentioned goal, different methods based on ideals are proposed to achieve this aim. These methods are more accurate than the previous methods. Hence it is very interesting in rough set context for removing the vagueness (uncertainty).

Rough set theory was introduced by Pawlak, in 1982, as a methodology to discover structural relationships within imprecise and uncertain data. This theory has been generalized using the idea of neighborhood systems to be more efficient to get rid of uncertainty and deal with a wide scope of practical applications. Motivated by this idea, in this work, we initiate novel generalized rough set models using the concepts of “maximal left neighborhoods and ideals.” Their basic features are studied and the relationships between them are revealed. The main merits of these models, as we prove, are first to preserve almost all major properties of approximation operators with respect to the Pawlak model. Second, they keep the monotonic property, which leads to an efficient evaluation of the uncertainty in the data, and third, these models enlarge the knowledge gotten from the information systems because they minimize the vagueness regions more than some previous models. We complete this manuscript by applying the proposed approach to analyze educational data and illustrate its role to improve the obtained classifications of objects and show the great performance of the present approach against other ones. Elucidative examples that support the obtained results are provided.

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.

Rough set theory was introduced by Pawlak in 1982 to handle imprecision, vagueness, and uncertainty in data analysis. It is dealing with vagueness (ambiguous) of the set by using the concept of the lower and upper approximations of objects based on an equivalence relation. The main idea of rough sets corresponds to study these approximations. So, in this paper, we generalize these approximations in the frameworks of topological spaces. The lower and upper approximations of Pawlak’s model are replaced by interior and closure notions of the topological space. The set approximations are defined using the new topological notions δβ _j -open sets and ⋀ _{β j} -sets. Such techniques open the way for more topological applications in rough context and help in formalizing many applications from real-life data. The current extension approximations are satisfied all properties of original rough set theory without any conditions or restrictions. Comparisons between the current approximations and the previous one are introduced and shown to be more general.

One of the most popular and important tools to deal with imperfect knowledge is the rough set theory. It starts from dividing the universe to obtain blocks utilizing an equivalence relation. To make it more flexibility and expand its scope of applications, many generalized rough set models have been proposed and studied. To contribute to this area, we introduce new generalized rough set models inspired by “maximal union neighborhoods and ideals.” These models are created with the aim to help decision‐makers to analysis and evaluate the given data more accurately by decreasing the ambiguity regions. We confirm this aim by illustrating that the current models improve the approximations operators (lower and upper) and accuracy measures more than some existing method approaches. We point out that almost all major properties with respect to rough set model can be kept using the current models. One of the interesting obtained characterizations of the current models is preserving the monotonic property, which enables us to evaluate the vagueness in the data and enhance the confidence in the outcomes. Moreover, we compare the current approximation spaces with the help of concrete examples. Finally, we show the performance of the current models to discuss the information system of dengue fever disease and eliminate the ambiguity of the medical diagnosis, which produces an accurate decision.