In this paper, it is introduced the notion of r -fuzzy ideal separation axioms T _i , i = 0, 1, 2 based on a fuzzy ideal I on a fuzzy topological space ( X , τ ). An r -fuzzy ideal connectedness related to the fuzzy ideal I is introduced which has relations with a previous r -fuzzy connectedness. An r -fuzzy ideal compactness related to I is introduced which has also relations with many other types of fuzzy compactness.

This paper introduces the notion of picture fuzzy modal topological multifunctions. These structures are grounded on novel picture fuzzy topological operators for closure and interior types, thereby utilizing the two standard picture fuzzy modal operators $ \square $ and $ \Diamond $. The paper discusses several fundamental properties of picture fuzzy multifunctions. Many types of structures are introduced, and some types of continuous multifunctions between picture fuzzy topological structures are discussed. The results indicate that some properties considered satisfactory in the intuitionistic fuzzy modal topological structures, as defined by Atanassov in 2022, are not fulfilled.

In this paper, lattice-valued analogues of T _i separation axioms for fuzzy bitopological spaces where i ∈ {0, 1, 2, 3, 4} are introduced, employing the notion of neighborhood filter of W. Gähler. In the case of L = {0, 1}, the axioms in question reduce to the classical separation axioms of general topology. Additionally, these lattice-valued generalizations have many properties of their corresponding crisp analogues. Also, showing that these separation axioms are good extensions in the sense of R. Lowen.

In this paper, we introduce the concept of fuzzy soft filters and study some of their properties. Also, we study the notion of convergence of fuzzy soft filters in fuzzy soft topological spaces. We prove the existence of product fuzzy soft filters.

Expanding on our research, this paper introduced novel generalizations of H ölder's and Minkowski's dynamic inequalities on diamond alpha time scales. Specifically, as particular instances of our findings, we replicated the discrete inequalities established when $ \mathbb{T = N} $. Furthermore, our investigation extended to the continuous case with $ \mathbb{ T = R} $, revealing additional inequalities that are both new and valuable for readers seeking a comprehensive understanding of the topic.