The concepts of supra soft somewhere dense closure, also known as SS-sd-closure, are first applied in this study to the problem of connectedness in supra soft topological spaces, or SSTSs. To be more specific, we use the concepts of SS-sd-components to describe a novel method of connectedness via SS-sd-sets, which we refer to as SSL-sd-connectedness, or supra soft locally sd-connectedness. SS-sd-hyperconnectedness, a different kind of connection via SS-sd-sets in SSTSs, is shown. We investigate the master characteristics of different kinds of connectedness. We found out that, SS-sd-hyperconnected spaces are equivalent to SS-sd-connected spaces, which distinguishes our concepts from their counterparts. Furthermore, we present two new forms of compactness, called SS-sd-compactness and SS-sd-Lindel\"{o}fness, which are based on the concepts of super soft sd-sets in the context of SSTSs. We get into a lot of detail about their primary features. In particular, we demonstrate that an SS-sd-compact (Lindel\"{o}f) soft set is the soft intersection of an SS-sc-set and an SS-sd-compact (Lindel\"{o}f) soft set. Furthermore, an SS-sd-compact (Lindel\"{o}f) SSTS with the soft finite (countable) intersection property also known as SFIP (SCIP) has been demonstrated in terms of its behaviour. Finally, we compare our novel soft versions of connectedness and compactness using SS-sd-sets with earlier research and add two topological charts to Figures 1 and 2 to illustrate the main ideas of this study. Concrete examples and counterexamples have verified that the arrows in these charts are non-reversible.

In today’s data‐driven world, we are constantly exposed to a vast amount of information. This information is stored in various information systems and is used for analysis and management purposes. One important approach to handle these data is through the process of clustering or categorization. Clustering algorithms are powerful tools used in data analysis and machine learning to group similar data points together based on their inherent characteristics. These algorithms aim to identify patterns and structures within a dataset, allowing for the discovery of hidden relationships and insights. By partitioning data into distinct clusters, clustering algorithms enable efficient data exploration, classification, and anomaly detection. In this study, we propose a novel centroid‐based clustering algorithm, namely, the morphological accuracy clustering algorithm (MAC algorithm). The proposed algorithm uses a morphological accuracy measure to define the centroid of the cluster. The empirical results demonstrate that the proposed algorithm achieves a stable clustering outcome in fewer iterations compared to several existing centroid‐based clustering algorithms. Additionally, the clusters generated by these existing algorithms are highly susceptible to the initial centroid selection made by the user.

This manuscript presents a new weaker version of supra septarian axioms based on supra $\epsilon$-open sets, along with its essential features, which are called supra-$\epsilon$-$T_{j}$-space, $j=0,1,2$, in the framework of supra topological spaces (or STSs). We give comprehensive explanations of each type of them, backed up by several examples and counterexamples that highlight the significance of our original approaches. We also provide a diagram that outlines these relationships. Additionally, we present the supra $\epsilon$-symmetric property and supra difference property and study their effects on these version of supra-$\epsilon$-septarian axioms. In especial, we show that the two concepts of supra-$\epsilon$-$T_{0}$-space and supra-$\epsilon$-$T_{1}$-space are the same for any STS that fulfills supra $\epsilon$-symmetric property. Finally, we study the supra topological and supra hereditary properties for each of the previously discussed approaches. In particular, we show that the property of being a supra-$\epsilon$-$T_{j}$-space, where $j=0,1,2$, is a supra-hereditary (topological) property.

An important mathematical way to deal with ambiguity and uncertainty in knowledge is rough set (RS) theory. It is believed that a grill is a necessary addition to this idea. Since it expands the approximate of RSs, it is a helpful technique for removing ambiguity and uncertainty. One of the key and important issues for developing rough sets, which subsequently aim to maximize the accuracy measure, is minimization of the boundary region (BR). One of the most practical and successful ways to accomplish this is with a grill. Thus, the goal of this work is to introduce novel grill-based approaches for rough sets (RSs). A few important aspects of these techniques are examined and illustrated to indicate that they produce accuracy measures that are higher and more significant than those of the previous methods. In the end, a medical application is shown to emphasize the need of using grills as instructed.

In recent years, fuzzy soft topology has emerged as a powerful approach that contributes significantly to advancing practical applications in uncertain environments, especially in information systems design, predictive modeling, and decision‐making. The structure of fuzzy soft topology, defined over fuzzy soft sets as a universal set instead of a crisp universal set, was introduced and discussed. This article aims to develop and present a comprehensive study of this structure in more depth by investigating new concepts, properties, and related findings, supported by illustrative examples. We also explore several forms and instances of this type of fuzzy soft topologies. The focus of this article is the exploration of new classes of separation and regularity axioms in the context of these types of fuzzy soft topology. We study these classes, establish their key properties, and reveal the interconnections between them. The relationships between the presented separation axioms and those in the induced crisp and soft topologies are examined. Moreover, we demonstrate that the separation and regularity axioms have the harmonic and hereditary properties. The new concepts, examples, and systematic investigation of the fuzzy soft separation and regularity axioms with their interrelationships achieved in this study contribute to the development of the theoretical foundation for the structure of topology defined over fuzzy soft sets. Moreover, our results not only deepen the understanding of this framework but also pave the way for its improved applications, especially in dealing with uncertain cases in real‐life problems.

In order to solve problems and provide practical solutions, researchers attempt to accurately describe societal difficulties and obstacles. An efficient method for handling complicated real-world data is rough set theory. The method finds confirmed and likely data from subsets using rough approximation operators. In order to increase accuracy while following Pawlak’s conventional approximation axioms, earlier research has created rough approximation models based on neighborhood systems. Based on cardinality rough neighborhoods and grills, we present new rough set notions in this study. These models are a suitable approach for a number of scenarios, including computational analysis, comparisons on medical datasets, real-world data analysis challenges, and classification accuracy metrics.We thoroughly examine the fundamental components of these ideas and clarify how they relate to one another as well as to earlier paradigms. Next, we describe boundary regions and assess the accuracy of the data using a topological technique. Additionally, we look at how well our models handle heart failure disease in certain individuals and come to the conclusion that the suggested rough set concepts improve upon the characteristics of the earlier approach spaces.Finally, we identify the shortcomings of the current concepts and show their advantages in terms of extending the verified information gleaned from subsets of data while preserving the key elements of Pawlak’s original paradigms that were destroyed by the models that went before them.