Employing a Generalization of Open Sets Defined by Ideals to Initiate Novel Rough Approximation Spaces with a Chemical Application (European Journal of Pure and Applied Mathematics, 2024)
A close similarity and analogy between rough set theory and topology is attributed to the corresponding behavior of lower and upper rough approximations with interior and closure topological operators, respectively. This relation motivates joint studies between topology and this theory. We endeavor by rough set theory to enlarge the knowledge we obtain from the information systems, for this reason, we apply the abstract concept of ideal structures to build new generalized approximation spaces with less vagueness. In the present work, we employ a novel type of nearly open sets in topology so-called ``${\mathcal{L}}$-${θβ}_{λ}$-open" with an ideal structure to introduce novel approximation spaces satisfying the desired properties concerning shrinking the boundary region of uncertainty and expanding the domain of confirmed information. We set up the fundamentals of the proposed rough paradigms and demonstrate their superiority over the preceding paradigms induced by some nearly open sets. Two algorithms are furnished to illustrate the way of specifying the family of ${\mathcal{L}}$-${θβ}_{λ}$-open sets and exploring whether a subset is ${\mathcal{L}}$-${θβ}_{λ}$-definable or ${\mathcal{L}}$-${θβ}_{λ}$-rough. Then, we put forward the concepts of rough membership relations and functions and uncover their core characterizations. Finally, we examine the proposed models to model a real situation in the Chemistry field and clarify how our models improve the outcomes of generalized approximation spaces over the previous models.
Keywords: Rough set; Topology; ideal; $\mathcal{L}$-${θβ}_{λ O}$-open set