Pure Mathematics for Theoretical Computer Science

A variety of uncertainty-handling frameworks—such as Fuzzy Sets,1 Hyperfuzzy Sets,2 Bipolar Fuzzy Sets,3 Neutrosophic Sets,4 Vague Set,5 Hesitant Fuzzy Sets,6, 7 Picture Fuzzy Sets,8 Soft Sets,9, 10 Rough Sets,11 and Plithogenic Sets12, 13—have been extensively studied for modeling and reasoning under vagueness and imprecision. A fuzzy set extends classical set theory by assigning each element a membership value in the unit interval [0, 1], thereby capturing partial inclusion.1 Neutrosophic Sets further generalize this idea by introducing three independent membership functions—truth, indeterminacy, and falsity—each mapping into [0, 1]. Many of these frameworks have been enriched by incorporating offset concepts, which permit membership degrees to take values beyond the unit interval. Similarly, D-numbers extend Dempster–Shafer belief functions by assigning to each subset B ⊆ X a mass D(B) ∈ [0, 1] with P B D(B) ≤ 1, thus accommodating incomplete uncertainty.14 In this work, we introduce and formally define four new constructs: D-OffNumber, D-OverNumber, D-UnderNumber, and Neutrosophic D-Number, and we investigate their mathematical foundations, structural properties, and interrelationships. The present study focuses exclusively on theoretical development, leaving potential applications—such as their integration into decision-making frameworks—for future research.

Fuzzy sets,20 rough sets,14 intuitionistic fuzzy sets,3 neutrosophic sets,15 soft sets,13 hesitant fuzzy set,17 plithogenic sets,16 and other uncertainty-handling frameworks have been the focus of intensive and ongoing research. Rough set theory provides a mathematical framework for approximating subsets through lower and upper approximations defined by equivalence relations, effectively capturing uncertainty in classification and data analysis.5, 10 Building upon these foundational concepts, further generalizations such as Hyperrough Sets8 and Superhyperrough Sets have been introduced. In this paper, we investigate the concepts of Neighborhood Hyperrough Sets and Neighborhood Superhyperrough Sets. These models extend the classical Neighborhood Rough Set framework by incorporating the structural richness of Hyperrough Sets and Superhyperrough Sets.

A weighted graph is a graph in which each edge is assigned a numerical value (weight), typically representing cost, distance, or intensity. In this paper, we revisit and further explore three generalizations of weighted graphs: the Hyperweighted Graph, the Superhyperweighted Graph, and the MultiWeighted Graph. These advanced structures were initially introduced in.10 Our objective is to enhance understanding and broaden awareness of their theoretical foundations and potential applications through renewed analysis and formal refinement

The main aim of this paper is extend the notion of S-extending fz-modules into LS-extending fz-modules and study this new notion. This lead us introduce and study other notions such as: purely semisimple, purely extending and purely y-extending fz-modules. Moreover, the relationships LS-extending fz-module with the various types.

In this paper, we introduce a class of weighted Orlicz spaces in the context of double coset spaces related to locally compact hypergroups in some way, which one can study that either these spaces are convolution algebras.