Extension of topological structures using lattices and rough sets

: This paper explores the application of Rough Set Theory in analyzing ambiguous data within complete information systems. The study extends topological structures using equivalence relations, establishing an extension of topological lattice within lattices. Various relations on topological spaces generate di ff erent forms of exact and rough lattices. Building on Zhou’s work, the research investigates rough sets within the extension topological lattice and explores the isomorphism between topology and its extension. Additionally, the paper investigates the integration of lattices and rough sets, essential mathematical tools widely used in problem-solving. Focusing on computer science’s prominent lattices and Pawlak’s rough sets, the study introduces extension lattices, emphasizing lower and upper extension approximations’ adaptability for practical applications. These approximations enhance pattern recognition and model uncertain data with finer granularity. While acknowledging the benefits, the paper stresses the importance of empirical validations for domain-specific e ffi cacy. It also highlights the isomorphism between topology and its extension, revealing implications for data representation, decision-making, and computational e ffi ciency. This isomorphism facilitates accurate data representations and streamlines computations, contributing to improved e ffi ciency. The study enhances the understanding of integrating lattices and rough sets, o ff ering potential applications in data analysis, decision support systems, and computational modeling.


Introduction
Lattices and rough sets serve as potent mathematical tools extensively applied in efficiently resolving numerous significant problems.Lattices find their utility in computer science and approximation spaces [26,47], while the genesis of rough sets can be traced back to Pawlak's introduction in the early eighties via the approximation space [49,51].Pawlak notably investigated a rough membership function [52].Various researchers have subsequently sought to generalize rough sets utilizing diverse binary relations [2,3,16,17,18,19,33,34,37,38,45,53,65,66].The abstract consideration of topological structures involves analyzing the universal set U using its family of open sets.Each space within this framework can be viewed as a lattice, denoted LA(U), associated with every subset of U. The study of these lattices is contingent upon the elements of LA(U), not the points in U. Initial explorations into topological extensions were conducted for Kolmogorov spaces [4,6,21,36].Diverse extension methodologies include employing ideals [43,63], algebraic concepts [22,41,44], matroids [40,57,60,69], graphs [1,14,15], and preordered topological structures [12,23,24,25,32,55,64].In the realm of topology, many researchers regard open sets as points in the topology lattice, with Pawlak's traditional approximation structure resembling a quasi-discrete topology where each subset is open [39,58,56].This concept introduces rough sets that may not necessarily be open.Thus, within a quasi-discrete topology, two distinct types of sets exist: Pawlak's upper and lower approximations manifest as open sets.In the topological lattice, these open sets correspond to the interior and closure of sets within a quasi-discrete topology.Furthermore, the lower (or upper) approximation in the topological lattice signifies the largest (or smallest) open set containing (or contained within) a given set, termed as the greatest lower bound (or least upper bound) in the lattice.Zhou and Hu [72] explored rough sets on a complete completely distributive lattice.
In fact, Zhou and Hu focused on the crisp power set of a universe, represented as an atomic Boolean lattice.Pawlak's rough sets were introduced, where equivalence classes were identified as elements of this power set lattice.The study then delved into lower and upper approximations based on the lattice order relation, leading to the exploration of natural ideals within a lattice.To extend this line of inquiry, they introduced a novel perspective by investigating rough sets from the standpoint of lattice theory.They took the lattice itself as the universe and explored the definition of lower and upper approximations within this lattice context.Building upon Järvinen's framework, Mordeson's approximation operators, and the work of Qi and Liu [54] on rough sets and generalized rough sets, their contribution lies in defining rough sets on a CCD lattice using an arbitrary binary relation.Unlike existing approaches that primarily focus on covers or partitions, they proposed a unified framework for the study of rough sets by employing a binary relation on a CCD lattice.This framework encompasses rough sets based on ordinary binary relations, rough fuzzy sets, and interval-valued rough fuzzy sets.The adoption of a binary relation introduces a new level of generality, allowing them to establish a connection between relations and partitions on CCD lattices.Furthermore, their paper established a pair of lower and upper approximation operators on CCD lattices based on this binary relation, offering a broader generalization of rough sets.They emphasized that their approach is not only applicable to CCD lattices but also extends the understanding of rough sets on Boolean lattices and power lattices.By demonstrating that the rough sets defined in [10] are special cases within their framework, they underscored the uniqueness and irreplaceability of their proposed rough sets.In summary, their manuscript contributes significantly to the field by introducing a comprehensive study of rough sets on lattices, specifically CCD lattices, through the innovative use of binary relations.We believe that this approach brings a fresh perspective to rough set theory, paving the way for new horizons in the understanding and application of rough sets, while others investigated them on general lattices [7,9,13,20,42,61,62].
Our research introduces lower and upper extension approximations within an extension lattice framework, exhibiting promise for integration into practical applications.These extension approximations demonstrate versatility, enabling their integration with specific applications in diverse domains.In data science and artificial intelligence, their capability to discern detailed relationships among elements could enhance pattern recognition and data analysis methodologies, contributing to more refined decision-making processes.Moreover, their finer granularity in defining rough and exact sets holds the potential for modeling and handling uncertain or imprecise data in practical applications, such as risk management systems or predictive modeling in financial markets.The advantages of our introduced extension approximations lie in their ability to capture more detailed relationships among elements, potentially leading to enhanced pattern recognition, data analysis, and decision-making processes.The finer granularity they provide in defining rough and exact sets facilitates more accurate and sophisticated modeling of uncertain or imprecise data, offering improved insights and solutions in real-world applications.Despite these advantages, it is essential to consider the limitations and disadvantages of existing methods that our proposed approach seeks to address.Some methods may lack the precision and detailed relationships offered by our extension approximations, making them less suitable for certain applications.Empirical validations and targeted studies are crucial steps toward demonstrating the efficacy and practical utility of our approach within specific domains.
In addition to our contribution, the paper explores the extension of topological structures through lattices, employing an equivalence relation to obtain an extension topological lattice.The comparison between the topological lattice and its extension is scrutinized, and various types of rough-bounded distributive lattices and their properties are investigated.The isomorphism between topology and its extension holds significant implications, particularly in practical applications involving data analysis, decision-making, and information retrieval systems.Understanding this isomorphism facilitates a more comprehensive characterization of relationships and structures within complex datasets.One practical significance lies in the realm of data representation and modeling.The isomorphism provides a bridge between the original topology and its extended form, enabling a seamless translation of concepts and relationships.This translation facilitates more accurate and efficient representations of complex data structures, which is invaluable in fields like pattern recognition, where precise data representations are critical.Furthermore, in decision-making processes, the isomorphism allows for a clearer understanding of relationships between different elements or features within datasets.This clarity aids in more informed decision-making by revealing underlying connections or similarities that might otherwise remain obscured without the isomorphic mapping.Moreover, the isomorphism between topology and its extension has implications in the realm of computational efficiency.It can streamline algorithms and computations by leveraging the correspondences between the original topology and its extended version.This streamlined approach could enhance the efficiency of various computational processes, such as data retrieval or analysis.In essence, the isomorphism between topology and its extension offers practical applications in data representation, decision-making, and computational efficiency.Its ability to bridge the gap between different topological structures enriches our understanding of complex datasets and enhances the efficacy of algorithms and processes utilized in various real-world applications.Furthermore, the paper acknowledges the importance of fuzzy lattices in the context of rough set theory.While our current discussion primarily centers on the representation of topological structures using lattices and rough sets, we recognize the need to explore fuzzy lattices to address the complexities inherent in data with fuzzy relationships.This acknowledgment opens avenues for future research endeavors, aiming to provide a more comprehensive understanding of how rough set theory can accommodate and address the nuances of real-world data.

Basic concepts and properties
Some basic concepts and results on lattices are introduced and studied in [5,8,11,25,59].The collection of topologies on fixed set X is a partially ordered.
In other words, if every open set of τ 1 is an open set of τ2, then τ 1 is weaker than τ 2 .We recall the following definitions which are useful in the sequel.
It is bounded if it has zero as a smallest element and unit as a greatest element of LA.
Definition 2.4.[12] Let LA be a lattice.Then, Then, a filter is nonempty subsets closed under meet.Definition 2.5.[43] The atom of a lattice LA to be an element x ∈ U s.t.x covers 0 (i.e.x > 0) and there is no element a s.t.x > a > 0).If every element in LA − {0} is the join atoms, then LA is an atomic lattice.Furthermore, if LA is finite, then it is a finite atomic lattice.Definition 2.6.[35] Let CLA = (B, ⊆) be a complete atomic lattice, A(B) be a lattice of B and φ : A(B) → B be any function, for any element x ∈ B. The lower and upper approximation of x w.r.to φ are Ω(x) = ∨{a ∈ A(B) : φ(x) ≤ x} , and

Rough sets
Definition 2.8.[48] Let (U, Ω) be an approximation structure, where Ω be an equivalence relation on U and U/Ω = {[x] Ω : x ∈ U} are equivalence classes of Ω.Then, for any X ⊆ U, lower and upper approximation of X are defined by and Proposition 2.9.[68] Let Ω be a relation on U. The following hold: (L1) Ω(X) ⊆ X.

Methods
In this section, the lattice through an equivalence relation is introduced.This relation forms an exact lattice on its equivalence classes.In this case, the relation between exact lattices and rough lattices is discussed.
The extension topology of quasi discrete topology (maybe for any topology) is established using rough sets.Moreover, for each G, H ∈ τ Ω s.t.G ≤ H, any element in τ Ω S has the form P = G ∪ (H ∩ P).So, if τ Ω = {U, ϕ, {a}, {b}, {a, b}, {c, d}, {a, c, d}, {b, c, d}}.For instance, if {a, d} is a rough set, then τ Ω S = {U, ϕ, {a}, {b}, {d}, {a, b}, {a, d}, {b, d}, {c, d}, {a, b, d}, {a, c, d}, {b, c, d}} is shown in Figure 2. It is noted that there is a homeomorphic between topologies of lattices which are extended by rough sets {a, c} and {a, d}.Remark 3.10.In the framework of (U, Ω), a subset is precisely categorized as an exact (or definable) set or a rough (or undefinable) set based on two distinct exact sets: the lower and upper approximation.While a rough set is defined approximately within (U, Ω), the exact set aligns with openness in Pawlak's approximation structure, forming a point within a lattice generated by an extension topology.Consequently, an open set is denoted as an exact point within a topological lattice, whereas any other set is termed a rough point.Essentially, any rough point can be defined by two exact points.For Proposition 3.11.Let ELA Ω (U) be a topology of an exact lattice and (U, τ Ω S ) be an extension topological structure of (U, τ Ω ).Then, ELA Ω (U) and (U, τ Ω S ) are equivalent.
Proof.Using Definitions 3.2 and 3.3, each of ELA Ω (U) and (U, τ Ω S ) is a topology.It is needed to prove that for each A ∈ ELA Ω (U), using Proposition 3.5, there is a unique point B ∈ (U, τ Ω S ) s.t.N(a, ELA Ω (U)) = M(b, (U, τ Ω S )) for a ∈ A and b ∈ B, respectively.Define a function f : ELA Ω (U) → (U, τ Ω S ) by f (a) = b.Thus, f is a bijective function from ELA Ω (U) onto (U, τ Ω S ), and for each a ∈ A, N(a, ELA Ω (U)) = N( f (a), (U, τ Ω S )).In particular, ∀ x ∈ U, N(β(x), ELA Ω (U)) = N( f (β(x)), (U, τ Ω S )), where β(x) is a basis element containing x. Finally, it is sufficient to find a homeomorphism f from Therefore, open sets in both of ELA Ω (U) and (U, τ Ω S ) in terms of a bijective function f are in a one to one correspondence.Hence, f is homeomorphism.□ Definition 3.12.Let (U, τ Ω S ) be an extension of (U, Ω).A lower and upper extension approximation for a rough set X are app S (X) = {Y ∈ U/Ω S : Y ⊆ X}.
The product of approximations of ELA Ω (U) has the form ELA Ω (U) ×ELA Ω (U) = {(app(X), app(X)) : X ⊆ U}.The location of each subset A in (U, Ω) is assigned with a function P : P(U) → Z + and is calculated by ).Similarly, the location of each subset A in (U, Ω S ) is assigned with a function P S : P(U) → Z + and is calculated by Note that: The introduced lower and upper extension approximations differ from the classical Pawlak approximation operators in the context of their construction within an extension lattice.The classical Pawlak approximation structure relies on exact sets-lower and upper approximations-to delineate between definable (exact) and undefinable (rough) sets within (U, Ω).However, the extension lattice enriches this paradigm by incorporating an equivalence relation-based extension of the Pawlak structure.Here, the lower and upper extension approximations transcend the classical approach by utilizing a generalized binary relation.This extension allows for a more nuanced depiction of relationships between elements, offering a refined characterization of rough and exact sets within a topological lattice.
When app(X) = app(X) = X, the scenario emerges where sup{Y ∈ ELA(U)} = inf{Y ∈ ELA(U)}, designating the rough set as exact.It's notable that the smallest rough set is (ϕ, ϕ), representing the empty set, while the greatest rough set is (U, U), signifying the entire universe U. Example 3.15 presents a concrete scenario where the extension lattice reveals a more detailed characterization of relationships among elements compared to the original lattice.The tables (Tables 1 & 2) with figures provide a visual representation of the sets and their approximations in both the original and extension lattices, highlighting the finer granularity achieved by our proposed approach.This example and comparisons substantiate the claim regarding the finer granularity of the extension lattice compared to the original lattice.
The assertion that the extension lattice offers increased granularity stems from its ability to capture more detailed relationships among elements than the original lattice.For instance, in the context of rough set approximations, the extension lattice, constructed through equivalence relations and generalized binary relations, allows for a more nuanced depiction of relationships between elements compared to the classical Pawlak approximation operators.Consider the scenario where the extension lattice incorporates an equivalence relation-based extension.This extension introduces a more extensive set of equivalence classes, resulting in a refined delineation of relationships among elements.Similarly, utilizing a generalized binary relation in defining lower and upper extension approximations enables a more detailed characterization of rough and exact sets within the lattice.While specific examples or comparisons illustrating the finer granularity of the extension lattice compared to the original lattice would enhance clarity, it's essential to emphasize that the increased detail and precision in representing relationships among elements are foundational to the claim of its finer nature.Further empirical studies and targeted comparisons between the extension and original lattices in specific applications would provide concrete instances demonstrating the enhanced granularity of the extension lattice.These comparative analyses would substantiate the claim and elucidate the practical implications of employing the extension lattice for more detailed data representations.We will strive to incorporate specific examples or comparative analyses in our work to bolster the claim regarding the finer nature of the extension lattice compared to the original lattice, thereby enhancing the comprehensibility and credibility of our findings.
Proof.Firstly, it is needed to prove that ELA(U) is lattice using meet and join on ELA(U) through union and intersection.Consider Clearly, ELA(U) ∈ ELA(U) × ELA(U).Now, it is sufficient to prove that ELA(U) is the supremum of all fixed point of (U, Ω).
Clearly, ELA(U) ∈ ELA(U) and it is the greatest lower bound of fixed points.Now, let G ⊆ U, then (P(G), ⊆) be a complete lattice.Let B = ({ELA(G), ⊆}) is a complete lattice of fixed points on ELA(G).

Conclusions
In this research, we explored the rich mathematical landscape of lattices and rough sets, essential tools in addressing various significant problems.Lattices, applied extensively in computer science and approximation spaces, and rough sets, originating from Pawlak's early work, have seen diverse generalizations utilizing different binary relations.Our focus extended to the abstract consideration of topological structures, where the universal set U was analyzed through its family of open sets, forming lattices denoted as LA(U).Topological extensions were explored for Kolmogorov spaces, employing methodologies such as ideals, algebraic concepts, matroids, graphs, and preordered topological structures.Zhou and Hu's exploration of rough sets on a complete completely distributive lattice provided a foundational contribution.Their innovative approach considered the lattice itself as the universe, defining rough sets on a CCD lattice using an arbitrary binary relation.Unlike previous methods focusing on covers or partitions, they proposed a unified framework, demonstrating the generality of their approach for ordinary binary relations, rough fuzzy sets, and interval-valued rough fuzzy sets.The significance of our manuscript lies in introducing a comprehensive study of rough sets on lattices, through the novel use of binary relations.We believe this approach offers a fresh perspective to rough set theory, opening new avenues for understanding and applying rough sets.Moreover, our research introduces lower and upper extension approximations within an extension lattice framework, showcasing promise for practical applications.These extension approximations exhibit versatility, offering integration potential in diverse domains such as data science and artificial intelligence.The finer granularity in defining rough and exact sets could enhance pattern recognition, data analysis methodologies, and decision-making processes, particularly in handling uncertain or imprecise data.While our approach presents several advantages, empirical validations and targeted studies are crucial steps to demonstrate its efficacy within specific domains.Future research endeavors could explore fuzzy lattices to address the complexities of data with fuzzy relationships, providing a more comprehensive understanding of how rough set theory adapts to real-world data nuances.Additionally, our investigation into the extension of topological structures through lattices and equivalence relations has practical implications.The isomorphism between topology and its extension facilitates more accurate data representation, decision-making processes, and computational efficiency.This bridging of different topological structures enriches our understanding of complex datasets and enhances the efficacy of algorithms and processes in real-world applications.In conclusion, our study contributes both theoretically and practically to the field of rough set theory, lattice theory, and topological structures.We believe that the insights gained from this research will stimulate further exploration, leading to advancements in various applications and domains.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Definition 3 . 7 .
Let (U, Ω) be an approximation structure.A minimal equivalence class for each a ∈ U is given by N(a) = {Y ∈ U/Ω : ∀a ∈ Y}.Definition 3.8.Let (U, Ω) be a approximation structure with |U| = n s.t n be finite and |U| be the cardinality of U. A minimal basis of (U, Ω) is defined by B N = {N(a i ) : a i ∈ U}.

Figure 1 .
Figure 1.Lattice on U ={a, b, c, d} and its topological extension

Figure 2 .
Figure 2.An extension lattice using a rough set {a, d} instance, if a, b ∈ LA(U) such that (a ≤ b), five cases exemplify their relation in the extension lattice: (i) Any rough point c LA(U) such that a ≤ c ≤ b.This establishes a ≤ c and c ≤ b. (U, Ω) (ii) If a ≤ b, then a ∨ b = b and a ∧ b = a in both lattices and their extensions.(iii) For a ≤ c ≤ b, a ∨ c = c, a ∧ c = a, c ∨ b = b, and c ∧ b = c.This configuration is termed a rough point approximation.(iv) If a ∨ b = a ∨ (c ∨ b) = (a ∨ c) ∨ b = c ∨ b = b, it denotes an upper rough point approximation.(v) When a ∧ b = (a ∧ c) ∧ b = a ∧ (c ∧ b) = a ∧ c = a, it signifies a lower rough point approximation.Thus, a ≤ b in the extending lattice, and the rough pair of any rough point lies within (a, b) such that a ≤ c ≤ b.

Figure 5 .
Figure 5. Rough lattice of a total undefinable rough set

Table 1 .
. □ Example 3.15.Let U = {a, bc} with U/Ω = {{a}, {b, c}}.Lower approximation, upper approximation and the location P for every A ⊆ U are given and computed in Table1and Figure3.If X = {a, b} is a rough set, then the extension of the approximation structure is τ Ω S = {U, ϕ, {a}, {b}, {a, b}, {b, c}}.In addition, the extension lower approximation and extension upper approximation are obtained and the location P S for every A ⊆ U is determined in Table2and Figure4.Lower and upper approximations and their locations.
F app (H app )}.So, F app (H app ) ⊆ H app .Since H app ⊆ F app (H app ), then F app (H app ) =H app .This means that H app is a fixed point.Similarly, H app is also fixed point.By the same manner, consider ELA By assumption, F is an increasing function, we get F(X) ⊆ F(H app ) and F app (X) ⊆ F app (H app ).Since X is a fixed point, i.e.F app (X) = X, then X ⊆ F app (H app ) and so U U = X {F app (H app ) : X ⊆