Essential Corrigenda to "Generalized approximation spaces generation from $\mathbb{I}_{j}$-neighborhoods and ideals with application to Chikungunya disease"
Rodyna A. Hosny, Naglaa M. Madbouly, Mostafa K. El-Bably
TL;DR
The corrigendum addresses errors in the prior work on generalized approximation spaces generated from $\mathbb{I}_{j}$-neighborhoods by offering corrected definitions, proofs, and illustrative counterexamples. It develops a rigorous framework for $\mathbb{I}^{\mathcal{K}}_{j}$-neighborhoods within an $\mathbb{I}$-$G$ approximation space, derives corrected relationships among $\mathbb{I}^{\mathcal{K}}_{j}(s)$, $\omega_{j}(s)$, and $\rho_{j}(s)$ under various properties of the relation $R$ and the ideal $\mathcal{K}$, and provides revised theorems (e.g., $t3$, $t4$) and lemmas (e.g., $l1$) along with updated tables. These corrections enhance the theoretical reliability of neighborhood-based rough set methods and clarify the construction of topologies from neighborhoods, with implications for data analysis and potential medical applications. The work strengthens the mathematical foundations of $\mathbb{I}$-generalized approximations and offers corrected pathways for future research.
Abstract
In the work "Generalized approximation spaces generation from $\mathbb{I}_{j}$-neighborhoods and ideals with application to Chikungunya disease, published in \emph{AIMS Mathematics}, \textbf{9}(4) (2024), 10050$-$10077," Al-Shami and Hosny introduced a novel approach for generating generalized neighborhoods through $\mathbb{I}_{j}$-neighborhoods with ideals, subsequently deriving new methods for generalized approximations based on these neighborhoods. However, several errors have been identified in their results, concepts, and methods, along with significant inaccuracies in the provided examples and comparison tables. This paper aims to address and correct these errors, providing counterexamples to illustrate the inaccuracies in the proposed results. Furthermore, we correct several flawed proofs and present the revised formulations of these results and concepts. Additionally, we offer properties and clarifications to enhance the understanding of these concepts.