Browsing by Author "Demiralp, S."

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New Insights into Rough Set Theory: Transitive Neighborhoods and Approximations
(Multidisciplinary Digital Publishing Institute (MDPI), 2024) Demiralp, S.
Rough set theory is a methodology that defines the definite or probable membership of an element for exploring data with uncertainty and incompleteness. It classifies data sets using lower and upper approximations to model uncertainty and missing information. To contribute to this goal, this study presents a newer approach to the concept of rough sets by introducing a new type of neighborhood called j-transitive neighborhood or j-TN. Some of the basic properties of j-transitive neighborhoods are studied. Also, approximations are obtained through j-TN, and the relationships between them are investigated. It is proven that these approaches provide almost all the properties provided by the approaches given by Pawlak. This study also defines the concepts of lower and upper approximations from the topological view and compares them with some existing topological structures in the literature. In addition, the applicability of the j-TN framework is demonstrated in a medical scenario. The approach proposed here represents a new view in the design of rough set theory and its practical applications to develop the appropriate strategy to handle uncertainty while performing data analysis.
On Strong (i,j)-Semi∗-Gamma-Open Sets in Ideal Bitopological Space
(2024) Bukhatwa, I.; Demiralp, S.
In this study, we introduce the concepts of $(i,j)$-semi$^{*}$-$Gamma $-open sets within the context of ideal bitopological spaces. This concept is demonstrated to be weaker than the established the notion of $(i,j)$-semi-$Gamma $-open sets. Subsequently, we define strong $(i,j)$-semi$^{*}$-$Gamma $-open sets in ideal bitopological spaces, elucidating some of their essential characteristics. Furthermore, leveraging this newly introduced concept, we establish the notions of strong $(i,j)$-semi$^{*}$-$Gamma $-interior and strong $(i,j)$-semi$^{*}$-$Gamma $-closure.
Topologically indistinguishable relations and separation axioms
(2024.01.01) Demiralp, S.; Al-shami, T.M.; Abushaheen, F.A.; El-latif, A.M.A.
This study focuses on defining separation axioms for sets without an inherent topological structure. By utilizing a mapping to relate such sets to a topological space, we first define a distinguishable relation over the universal set with respect to the neighborhood systems inspired by a topology of the co-domain set and elucidate its basic properties. To facilitate the way of discovering this distinguishable relation, we initiate a color technique for the equivalence classes inspired by a given topology. Also, we provide an algorithm to determine distinguishable members (or objects) under study. Then, we establish a framework for introducing separation properties within these structureless sets and examine their master characterizations. To better understand the obtained results and relationships, we display some illustrative instances.