In this work, we introduce the concept of (∈,∈∨(k^*,𝔮_k)) -fuzzy quasi-ideals in ordered semigroups. We provide several equivalent conditions of these fuzzy quasi-ideals and demonstrate that the intersection of any collection of these fuzzy quasi-ideals is of the same type, but their union is not. We also discuss how these novel fuzzy quasi-ideals connect to previously studying fuzzy quasi-ideal notions in the literature. Additionally, we provide characterizations of (∈,∈∨(k^*,𝔮_k)) -fuzzy quasi-ideals in terms of (k^*,k) -upper part of fuzzy sets. We further introduce the notion of completely semiprime (∈,∈∨(k^*,𝔮_k)) -fuzzy quasi-ideals and establish several equivalent conditions for them in terms of (k^*,k) -upper parts of fuzzy sets and ordered fuzzy points. Finally, we characterize the left and right regularities of ordered semigroups by using completely semiprime (∈,∈∨(k^*,𝔮_k)) -fuzzy quasi-ideals.

We study natural smooth structures on Hausdorff locally convex spaces from the point of view of diffeology and their application to nonlinear operations on distributions. We show that a Hausdorff locally convex space is convenient if and only if the canonical map from the space to its internal tangent space is an isomorphism, thereby giving a geometric characterization of Mackey completeness. We also examine the behavior of these diffeologies under completion, duality, and inductive limits. As an application, we construct the space of microlocally multipliable distributions as a diffeological colimit of wavefront-restricted distribution spaces and prove that the induced multiplication map into the space of distributions is smooth. The construction uses only the standard wavefront-set criterion for multiplication and illustrates how diffeological colimits encode domains of partially defined nonlinear operations. This yields a diffeological framework for Hörmander-admissible multiplication beyond the classical manifold setting.

In this paper, we investigate anti-fuzzy primary ideals in the setting of $(m,n)$-near rings. We study their basic properties and present several characterizations that describe their algebraic structure. The relationships between anti-fuzzy primary ideals and classical primary ideals in $(m,n)$-near rings are examined, as well as their connections with other classes of anti-fuzzy ideals. Moreover, we analyze the behavior of anti-fuzzy primary ideals under $(m,n)$-near ring homomorphisms and prove that, under suitable conditions, their homomorphic images and preimages preserve primary-type properties. These results extend existing theories of fuzzy and anti-fuzzy ideals to the broader framework of $(m,n)$-near rings and provide further insight into their ideal structure.

In this paper, first the fuzzy set and i-(m,n)-near ring then fuzzy subgroups, fuzzy (m,n)-subnear ring and finally fuzzy ideals are defined. We state and prove theorems in these cases. Also in the next chapter, normal fuzzy subnear ring expressed and theorems about normal fuzzy subnear rings is expressed and proved.

Advanced technologies are experiencing significant transformation through AI (Artificial Intelligence), where mathematical foundations serve as the backbone of classical and modern AI models. Despite rapid advancements and widespread adoption, there is a lack of a unified framework for core AI components, comprising building blocks, governing equations, parameters, evaluation criteria, benchmarks, performance metrics, and objective functions across technological domains. In this comprehensive survey, we address this gap in the areas of energy, renewable energy and water, smart buildings and cities, the environment and climate change, hydrogen and hydrogen fuel cells, and cross-sector advanced technologies, including robotics and autonomous systems, computer vision, finance, and industrial manufacturing, with systematic intercomparison and benchmarking. In addition, we conduct a taxonomy-based analysis with a methodological focus on AI models, their underlying parameters and governing equations, as well as their pros, cons, trade-offs, comparative analyses, and directions for future development. This survey consolidates these elements into a structured reference that defines key requirements for AI development in high-tech sectors and provides a forward-looking roadmap to foster innovation beyond current technological infrastructures.

In this article, we first define hyperstructures known as Krasner hypermodules. Then, the concept of topological Krasner hypermodules is explored, examining their fundamental properties and the notion of continuous mappings that exist between such topological hyperstructures. Next, the concept of Hausdorff topology is introduced and its relation to Krasner hypermodules is examined. The relationship between locally compact Krasner hypermodules and the role of open neighborhoods in their topological structure is then analyzed. Several theorems are presented and proven to clarify these relationships. By applying relative topology to subhypermodules, their associated properties are analyzed. In other words, the aim is to use specific topologies to identify the various substructural features of this type of hypermodule. Additionally, the quotient topology induced by the theta(& lowast;)-relation on the Krasner hypermodule is investigated to understand how this relation affects the topological structure of the hypermodule. Finally, it is shown that the topological Krasner hypermodule induced by tau theta, the finest and strongest topology on it, ultimately forms a module.

This paper introduces and systematically develops the notion of fuzzy primary ideals in the framework of (m,n)-near rings. We begin by formulating a precise definition of fuzzy primary ideals and then clarify the concept through illustrative examples. Subsequently, we establish several fundamental theorems that provide a rigorous characterization of their structural properties and algebraic behavior. These results extend fuzzy ideal theory to the broader setting of (m,n)-near rings and contribute new insights into the underlying structure of fuzzy algebraic systems.

Pythagorean fuzzy soft set (PFSS), an extension of intuitionistic fuzzy soft set (IFSS), proves instrumental in managing ambiguity within numerous real-life scenarios. When applied to graph theory, these sets introduce a range of novel concepts, such as Pythagorean fuzzy soft graphs (PFSGs), complete Pythagorean fuzzy soft graphs, strong Pythagorean fuzzy soft graphs (Str-PFSGs), and the self-complement of Pythagorean fuzzy soft graphs (Self-Comp PFSGs). Our exploration involves detailing diverse construction methods and delving into their associated properties to gain a comprehensive understanding of their applicability and significance. Finally, some operations related to the above concepts are stated and proved.

The concept of ideals in algebraic structures plays an important role in studying their structure. In this paper, we enrich an algebraic structure, which is a generalization of ordered groups called ordered semigroups, by using the more general form of fuzzy subsemigroups and fuzzy (generalized) bi-ideals. In this aim, the idea of (∈,∈∨(k^*,q_k)) -fuzzy subsemigroups in ordered semigroups is firstly defined. Then, we show that every fuzzy subsemigroup is a (∈,∈∨(k^*,q_k)) -fuzzy subsemigroup but the converse is not true in general. An equivalent condition for a fuzzy set to be a (∈,∈∨(k^*,q_k)) -fuzzy subsemigroup is provided. Additionally, the notions of (∈,∈∨(k^*,q_k)) -fuzzy generalized bi-ideals and (∈,∈∨(k^*,q_k)) -fuzzy bi-ideals in ordered semigroups are also defined. Several conditions are given for (∈,∈∨(k^*, q_k)) -fuzzy generalized bi-ideals to be (∈,∈∨(k^*,q_k)) -fuzzy bi-ideals. We prove that each fuzzy (generalized) bi-ideal is (∈,∈∨(k^*,q_k)) -fuzzy (generalized) bi-ideal; however, the converse is not true, as demonstrated by an example. Furthermore, we provide an equivalent condition for the (∈,∈∨(k^*,q_k)) -fuzzy (generalized) bi-ideal. Moreover, we characterize (∈,∈∨(k^⋆ ,q_k)) -fuzzy (generalized) bi-ideals using level subsets, (∈∨ (k^⋆ ,q_k)) -level subsets, and characteristic functions. Finally, by applying the ideal of the (k^⋆ , k) -upper part of fuzzy sets, more comprehensive characterizations of (∈,∈∨(k^⋆ ,q_k)) -fuzzy (generalized) bi-ideals are studied.

This study investigated the cubic intuitionistic fuzzy set of TM-algebra as a generalization of the cubic set. First, a cubic intuitionistic ideal and a cubic intuitionistic T-ideal are defined, followed by a discussion of their properties. Furthermore, the level set of a cubic intuitionistic TM-algebra is defined, and the relationship between a cubic intuitionistic level set and the cubic intuitionistic T-ideal is established. A novel definition of a cubic intuitionistic set under homomorphism is proposed, and several significant results are demonstrated.

Algebraic homomorphisms are essential mathematical structures that sustain operations across algebraic systems such as groups, rings, and fields. These mappings not only preserve the validity of algebraic operations but also make it easier to investigate structural similarities and equivalences across distinct algebraic entities. In this article, we establish the group isomorphism under the complex intuitionistic fuzzy set, an extended form of the complex fuzzy set that adds the complex degree of non-membership functions, which plays a significant role in the decision-making process. The complex algebraic structure provides effective tools for understanding complex phenomena. We discuss the more intricate features of homomorphism and isomorphism in the framework of a complex intuitionistic fuzzy set. In addition, we introduce the complex intuitionistic fuzzy normal subgroups. We establish the relationship between two complex intuitionistic fuzzy subgroups and analyze of complex intuitionistic fuzzy isomorphisms among these subgroups, proving the important theorems. Furthermore, we establish examples to explore the concept of complex intuitionistic fuzzy subgroups.

Algebraic systems are better understood through their subsets. On the other hand, fuzzy sets help in dealing with ambiguities. The combination of these concepts led to founding the theory of fuzzy algebraic structures. This paper studies semigroups through their antiideals and bi-antiideals and fuzzifies them. First, it investigates the properties of antiideals and bi-antiideals of semigroups. Then it fuzzifies these concepts to fuzzy antiideals and fuzzy bi-antiideals of semigroups. Finally, it studies these new concepts and establishes a relationship between them and antiideals (bi-antiideals) of semigroups through level sets.

In this paper we present a new generalization of CS, ECS and CCLS- modules. If each “cec-closed submodule “in a module M is a “direct summand“, then M is referred to be CECS. It was demonstrated that every ECS and CCLS-module is generalized by the CECS property. We look at modules M that allow one to lift all homomorphism from a cec-closed submodule of M to M. Despite this, certain modules have some characteristics in common with CECS modules.

In this study we define the radical of the Krasner hypermodules in the subcategory RShmod, then we use short exact sequences in homological algebra for Krasner hypermodules. Besides, by studying the concept of tau-supplements in module theory we will generalize it to the Krasner R-hypermodules by using short exact sequences and a subcategory of RS hmod.

The construction of circuits formed by reduced quadratic irrational numbers (RQINs) under the action of Mobius groups has attracted growing attention due to their deep algebraic structure and wide range of applications. Such orbits and circuits play a significant role in modern cryptographic systems, particularly in the design of robust substitution boxes (S-boxes), secure data encryption protocols, and image processing algorithms. The main objective of this novel study is to classify the types of H-circuits with different lengths contained in H-orbits eta H, where eta is a RQIN and H is a Hecke group. For a specific eta, the circuits of different lengths may contain eta,eta,-eta and -eta either lie in one circuit or a different circuit of the same orbit. Also, we discuss the behavior of RQINs in the coset diagrams under the action of group M=u,f:u2=f6=1. Furthermore, the general form of reduced numbers in specific orbits under certain circumstances on prime p is investigated by applying the concept of congruence. Finally, special attention is given to the classification of M-circuits of length two in M-orbits eta M.

We introduce primitive hyperideals of a hyperring R and show relations with R itself, and with maximal and prime hyperideals of R. We endow a Jacobson topology on the set of primitive hyperideals of R and study topological properties of the corresponding hyperstructure space.

This work is intended as an attempt to motivate a novel model of disordering discrete-time quantum walk in a one-dimensional lattice of integers. We construct such a model over max-plus algebra and give the notion of coin operator of the disordered discrete-time quantum walk in a one-dimensional lattice ℤ to derive a very complicated decision matrix. Furthermore, we investigate the properties of the decision matrix for each state and prove some results for the disordered quantum walk over max-plus algebra that are similar to the conserved quantity of the conventional disordered quantum walk.

In this paper we verify a connection between fuzzy sets, biological inheritance and hyperstructures. We analyse the second type Supplementary of non-Mendelian inheritance also simple inheritance and determine the sequences of join spaces and fuzzy sets associated to each of its types, focusing on the calculation of their lengths that is called the fuzzy grade of H.