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An essential and quite different class of functions is the triangular norm and its related operators (uninorms, nullnorms, and associative copulas). They are used widely in many disciplines, including fuzzy set theory, probability and statistics, decision sciences, and others. This paper proposes the notion of hybrid Ξ -norm and defines the concepts of Ξ -hybrid ideals, Ξ -hybrid h -ideals in a hemiring . Some equivalent conditions are obtained for a hybrid structure to be a Ξ -hybrid left h -ideal, and it is proved that every imaginable Ξ -hybrid left h -ideal of hemiring is a hybrid left h -ideal, but not conversely by an example. [ABSTRACT FROM AUTHOR]

The fuzzy set is a fantastic tool for expressing hesitancy and dealing with uncertainty in real-world circumstances. Soft set theory has recently been developed to deal with practical problems. The soft and fuzzy sets were combined by Jun et al. to generate hybrid structures. The idea of hybrid ideals on a distributive lattice is discussed in this work. The relation between hybrid congruences and hybrid ideals on a distributive lattice is also examined. In addition, the product of hybrid ideals and its numerous results are discussed. [ABSTRACT FROM AUTHOR]

The fuzzy set is highly beneficial for expressing people's hesitations in their everyday lives, and it is a great tool for dealing with uncertainty, which can be described precisely and perfectly from the decision-maker's point of view. Soft set theory has been developed in recent years to address realworld issues. Jun et al. merged fuzzy and soft sets to produce hybrid structures. Hybrid structures are soft set and fuzzy set speculations. The concept of hybrid ideals in near-subtraction semigroups is introduced in this paper, and their equivalent results are obtained. Additionally, we demonstrate the concept of hybrid intersection. Moreover, we define the concept of homomorphism of a hybrid structure in a near-subtraction semigroup. [ABSTRACT FROM AUTHOR]

Models of universe problems are brimming with complexities and uncertainties in almost every field of study, including engineering, mathematics, medical sciences, computer science, physics, management sciences, artificial intelligence, and operations research. To address these uncertainties, various theories have been developed, including probability, rough sets, fuzzy sets, soft ideals, and neutrosophic sets. Neutrosophic set theory is the focus of this paper. In this paper, we introduce the notions of neutrosophic N-filters and neutrosophic N-bi-filters in a semigroup and investigate several properties. Moreover, the relations of prime biideal subset and prime neutrosophic N-bi-ideal structure; neutrosophic N-bi-filter and neutrosophic N-bi-ideal structure; left (resp., right) filter and neutrosophic N-left (resp., right) filter; neutrosophic N-left(resp., right) filter and prime neutrosophic N-left (resp., right) ideals in semigroups are discussed. Finally we prove that: let X be a semigroup and XN be any neutrosophic structure. Then XN is a neutrosophic N-bi-filter of X if and only if XNc is a prime neutrosophic N-bi-ideal of X. [ABSTRACT FROM AUTHOR]

In general, ordered algebraic structures, particularly ordered semigroups, play an important role in fuzzification in many applied areas, such as computer science, formal languages, coding theory, error correction, etc. Nowadays, the concept of ambiguity is important in dealing with a variety of issues related to engineering modeling problems, network theory, decision-making problems in real-life situations, and so on. Several theories have been developed by various researchers to overcome the difficulties that arise from uncertainty, including fuzzy sets, intuitionistic fuzzy sets, probability, soft sets, neutrosophic sets, and many more. In this paper, we focus solely on neutrosophic set theory. In ordered semigroups, we define and investigate the properties of neutrosophic ℵ-ideals and neutrosophic ℵ-interior ideals. We also use neutrosophic K-ideals and neutrosophic ℵ-interior ideals to characterize ordered semigroups. [ABSTRACT FROM AUTHOR]

Human endeavours span a wide spectrum of activities which includes solving fascinating problems in the realms of engineering, arts, sciences, medical sciences, social sciences, economics and environment. To solve these problems, classical mathematics methods are insufficient. The real-world problems involve many uncertainties making them difficult to solve by classical means. The researchers world over have established new mathematical theories such as fuzzy set theory and rough set theory in order to model the uncertainties that appear in various fields mentioned above. In the recent days, soft set theory has been developed which offers a novel way of solving real world issues as the issue of setting the membership function does not arise. This comes handy in solving numerous problems and many advancements are being made now-a-days. Jun introduced hybrid structure utilizing the ideas of a fuzzy set and a soft set. It is to be noted that hybrid structures are a speculation of soft set and fuzzy set. In the present work, the notion of hybrid ideals of a near-ring is introduced. Significant work has been carried out to investigate a portion of their significant properties. These notions are characterized and their relations are established furthermore. For a hybrid left (resp., right) ideal, different left (resp., right) ideal structures of near-rings are constructed. Efforts have been undertaken to display the relations between the hybrid product and hybrid intersection. Finally, results based on homomorphic hybrid preimage of a hybrid left (resp., right) ideals are proved. [ABSTRACT FROM AUTHOR]