Correctly determining a company's market worth during an entire year or a certain period presents a difficulty to decision-makers. In the case of the merger of companies, the need performs heavier when both the companies' owners are attracted to establishing a fair price at the optimal time to merge. The effectiveness of representing, connecting and manipulating both uncertainty and periodicity information becomes highly required. Hence, study and nhance some properties and conditions of the algebraic structure of complex hesitant fuzzy graphs. Therefore, the degree of composition between two complex hesitant fuzzy graphs is proposed. Also, the formal definitions of union, joint and complement are presented to be covered in the realm of complex hesitant fuzzy graphs. A real-life application is illustrated to show the relation between vertices and edges in the form of complex hesitant fuzzy graphs.

In the ideal topological spaces (X, tau, I) and (Y, J, sigma), we define (I, J)-irresolute function, weakly-(I, J)-irresolute, strongly-(I, J)-irresolute functions, semi-I-open sets and quasi-H-closed spaces modulo ideal I. We discuss properties of (I, J)-irresolute function mapping connected, closed, compact ideal spaces modulo I and semi-I-isolated points. (I, J)-Homeomorphisim on presemi-I-open sets and closure properties of quasi-H-closed spaces and semi-I-open subsets are explored in this paper.

In this research work, we introduce the concept of countably α-compact spaces in an ideal topological space (X,τ,I) and study further properties of α-continuous functions. We prove that α-continuous function mapping a countably α-compact ideal topological space (X,τ,I) to the ideal space (Y,σ,J) is an α-closed subset of the Cartesian product (X×Y,τ×σ,I×J). We showed that the countably α-compact ideal space (X,τ,I) with weight ∣X∣≥ ℵ₀ is the α-continuous image of a closed subspace of the cube D^({ℵ₀}) and illustrate that the α-continuous function f:(X,τ,I)→(Y,σ,J) where Y is countably α-compact can be extended over its domain under some constraints. Moreover, α-pseudocompact is defined in an ideal topological space (X,τ,I) and we proved that countably α-pseudocompactness is not hereditary with respect to α-closed sets and we showed that countab α-pseudocompactness is not finitely multiplicative. We find that if the ideal topological space (X,τ,I) is Tychonoff, (Y,σ,J) is countably α-pseudocompact, and f:(X,τ,I)→(Y,σ,J) is α-continuous perfect function modulo (I,J), X is countably α-psuedocompact.

In the final years of the 20th century, the notion of generalized topological spaces was introduced, marking a significant shift in the field of topology. This paper focuses on a subset of on a non-empty set that is closed under arbitrary unions, defining a generalized topology and subsequently a generalized topological space (GTS) denoted by . Within this framework, we explore the concept of Noetherian generalized topological spaces and delve into the properties of closed subsets within the Noetherian GTS. The investigation reveals that subspaces of a Noetherian GTS , with the induced topology, inherit the Noetherian property and exhibit finitely many non-empty irreducible components. Furthermore, the study extends to the analysis of hereditary properties, regular , , irreducible closed subsets, and the product properties of closed subsets under continuous functions. We also establish the closure property of finite unions in Noetherian GTS and clarify the homeomorphic nature of Noetherian GTS to itself.

This paper examines the properties of α-continuous functions modulo (J, I) that map a countably α-compact ideal space (X, τ, J) to an ideal space (Y, σ, I), where Y is an α-closed subset of the Cartesian product (X × Y, τ × σ, J × I). It is shown that if (X, τ, J) has a weight of at least ℵ₀, it is the α-continuous image of a closed subspace of the cube D^({ℵ₀}). Additionally, an α-continuous function f: (X, τ, J) → (Y, σ, I), where Y is countably α-compact, can be extended under specific conditions. The concept of α-pseudocompactness is introduced in an ideal topological space (X, τ, J), and it is established that countably α-pseudocompactness is neither finitely multiplicative nor hereditary with respect to α-closed sets. Furthermore, it is proven that an α-continuous function modulo ideals mapping a Tychonoff ideal space to a countably α-pseudocompact space is perfect, and the Tychonoff space itself is countably α-pseudocompact.

In this paper, we introduce the concepts of D_α^p-open, D_α^p-closed subsets, pairwise-α-closed, pairwise-g-closed subsets, pairwise-strongly α-closed graph G(f) and strongly D_α^p-closed graph of bitopological spaces. We showed that each closed graph is D_α^p-closed. In addition, the concepts of D_α^p-continuous, open, and closed functions are defined, and the relations between τ_p-α, τ_p-g, and D_α^p-continuous functions are clarified. The fact that strongly D_α^p-closed graph is D_α^p-closed is illustrated. We studied when the graph G(f) is p-strongly closed and p-D_α-closed subsets of the bitopological space (X,τ₁,τ₂). Moreover, the notions of D_α^i-interior of a subset of X and D_α^i-closure are defined.