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Humanity's efforts are manifested in the creation of novel solutions to complex problems in diverse fields. Traditional mathematical methods fail to solve real-world problems due to their complexity. Researchers have come up with new mathematical theories like fuzzy set theory and rough set theory to help them figure out how to model the uncertainty in these fields. Soft set theory is a novel approach to real-world problem solving that does not require the membership function to be specified. This aids in the resolution of a wide range of issues, and significant progress has recently been made. After Jun et al. came up with a hybrid system that combined fuzzy and soft set concepts, many people came up with hybrid ideas in different algebraic structures. In this paper, we introduce the concepts of subsystem and strong subsystem of a hybrid finite state machine (HFSM) and investigate a portion of their significant properties. We also provide an example that shows that every subsystem does not need to be a strong subsystem. Additionally, we study the cyclic subsystem of HFSMs and also obtain their equivalent results and examples. Finally, we define the notions of homomorphism of subsystems and strong subsystems of HFSMs. [ABSTRACT FROM AUTHOR]

The concepts of Dokdo sub-hoops and Dokdo filters are introduced, and their properties are investigated. The relationship between Dokdo sub-hoops and Dokdo filters is discussed, and characterizations of Dokdo sub-hoops and Dokdo filters are established. [ABSTRACT FROM AUTHOR]

Fuzzy sets, soft sets, and their generalisations have always been important tools for mathematicians and researchers working with uncertainty. Jun proposed a hybrid structure that combined the concepts of a fuzzy set and a soft set. It should be noted that hybrid structures are a combination of soft set and fuzzy set speculation. Our aim is to explore the concept of hybrid ordered ideals and hybrid interior ideals in ordered semirings, as well as look at some of their related properties, which is one of the important aspects of this paper. In order to investigate the structure theory of hybrid ideals in ordered semirings, we define hybrid composition and hybrid addition. We also establish and characterise the regularity of ordered semirings in terms of hybrid structures. [ABSTRACT FROM AUTHOR]

The notion of (closed) Łukasiewicz fuzzy ideal is introduced, and several properties are investigated. The relationship between Łukasiewicz fuzzy subalgebra and Łukasiewicz fuzzy ideal is discussed, and characterization of a Łukasiewicz fuzzy ideal is considered. Conditions for a Łukasiewicz fuzzy subalgebra to be a Łukasiewicz fuzzy ideal are provided, and conditions for the 2-set, q-set and O-set to be ideals are explored. [ABSTRACT FROM AUTHOR]

Many uncertainties arise in real-world problems, making them impossible to solve using conventional approaches. Researchers all over the world have developed new mathematical theories like fuzzy set theory and rough set theory to better understand the uncertainties that occur in various fields. Soft set theory, which was recently introduced, offers a novel approach to real-world problem solving by removing the need to set the membership function. This is helpful in resolving a variety of issues, and much progress is being made these days. Recently, Jun introduced the concept of a hybrid structure, which blends the concepts of a fuzzy set as well as a soft set. In this paper, we define the hybrid k-sum and hybrid k-product of k-ideals of semiring and investigate their properties. We illustrate with an example that the hybrid sum and hybrid product of two k-ideals are not always hybrid ideals. We also describe semiring regularity constraints in terms of hybrid k-ideal structures. [ABSTRACT FROM AUTHOR]

We study the lattice structure of fuzzy A-ideals in an mv-module M (fai (M), symbolically) and show that it is a complete Heyting lattice and so the set of its pseudocomplements forms a Boolean algebra. In the sequel, the properties of fuzzy congruences in an mv-module are investigated and using them some structural theorems are stated and proved. Finally, it is proved that fai (M) can be embedded into the lattice of fuzzy congruences. [ABSTRACT FROM AUTHOR]

In general, models of universe problems in almost all fields, such as engineering, mathematics, medical sciences, computer science, physics, management sciences, artificial intelligence, and operations research, are practically full of complexities and include various types of uncertainties when dealing with them on numerous occasions. Different theories, such as probability, rough sets, fuzzy sets, soft ideals, and so on, have been created to deal with these uncertainties. An algebraic structure, AG-groupoid, is an intermediate structure between two types of structures: commutative semigroup and groupoid. This structure has a very close relationship to a commutative semigroup since a commutative AG-groupoid is always a semigroup. These structures have so many applications in flocks theory, geometry, topology, and many more. We explore several structural properties of an AG-groupoid by using hybrid structures in this paper. The main motivation behind this paper is to present the concepts of hybrid ideals, hybrid bi-ideals and hybrid interior ideals of an AG-groupoid and characterize AG-groupoid in terms of hybrid structures. Also, we show that the hybrid intersection and hybrid product structures will coincide under certain conditions. [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]

The concept of Sup-hesitant fuzzy p-ideal in BCK/BCI-algebras is introduced, and related properties are investigated. A characterisation of Sup-hesitant fuzzy p-ideal of BCK/BCI-algebras is provided. Relations between Sup-hesitant fuzzy p-ideal and Sup-hesitant fuzzy ideal are investigated. Conditions for a Sup-hesitant fuzzy ideal to be a Sup-hesitant fuzzy p-ideal are provided. Characterisation of p-semisimple BCI-algebra is considered. Extension property for Sup-hesitant fuzzy p-ideal is established. [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]

Rosenfeld [A. Rosenfeld, Fuzzy Graphs, Fuzzy Sets and Their Applications, eds. L. A. Zadeh, K. S. Fu and M. Shimura (Academic Press, New York, 1975), pp. 77–95.] defined the fuzzy relations on the fuzzy sets and developed the structure of fuzzy graph, as a graph with a membership degree (between zero and one) for the vertices and edges such that the membership degree of every edge is less than or equal to the minimum of the membership degree of its endpoints. Although this model of graph has many applications in the real life, it fails to solve a lot of problems, which we can use graph for its representation. This paper aimed to demonstrate a new type of graph with a membership degree (between zero and one) for the vertices and edges so that the membership degree of every edge becomes more than or equals the minimum of the membership degrees of its endpoints. This new type of graph is called inverse fuzzy graph "or" I-fuzzy graph, which can play a role in solving many problems which are not solved by fuzzy graph. [ABSTRACT FROM AUTHOR]

Using the concept of hesitant union (⋓), the notion of ⋓-hesitant fuzzy p- ideals is introduced, and their properties are investigated. Relations between ⋓-hesitant fuzzy ideals and ⋓-hesitant fuzzy p-ideals are considered. Conditions for the ⋓-hesitant fuzzy ideal to be a ⋓-hesitant fuzzy p-ideal are provided. The reduction property for ⋓-hesitant fuzzy p-ideals is established. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of how to define nearness BCK-algebra. Also, some properties of nearness BCK-algebras are investigated. [ABSTRACT FROM AUTHOR]

This paper consider the general forms of (α, β)-fuzzy left ideals (right ideals, bi-ideals, interior ideals) of an ordered semigroup, where α, β ∈ {∈λ qδ, ∈λ, ∧qδ, ∈λ v qδ} and ≠∈λ ∧qδ. Special attention is paid to (∈λ, ∈λ, v qδ)-left ideals (right ideals, bi-ideals, interior ideals) and some related properties are investigated. The characterization of regular ordered semigroups in terms of (∈λ, ∈λ vqδ)-fuzzy left (right) ideals, (∈λ, ∈λ vqδ)-fuzzy bi-ideals and (∈λ, ∈λ vqδ)-fuzzy interior ideals is also investigated. [ABSTRACT FROM AUTHOR]

Fuzzy soft positive implicative hyper BCK-ideal of types (≪,⊆, ⊆), (≪,≪, ⊆) and (⊆,≪,⊆) are introduced, and their relations are investigated. Relations between fuzzy soft strong hyper BCK-ideal and fuzzy soft positive implicative hyper BCK-ideal of types (≪, ⊆, ⊆) and (≪, ≪, ⊆) are discussed. We prove that the level set of fuzzy soft positive implicative hyper BCK-ideal of types (≪,⊆, ⊆), (≪,≪,⊆) and (⊆,≪,⊆) are positive implicative hyper BCK- ideal of types (≪,⊆, ⊆), (≪,≪,⊆) and (⊆,≪,⊆), respectively. Conditions for a fuzzy soft set to be a fuzzy soft positive implicative hyper BCK-ideal of types (≪, ⊆, ⊆), (≪,≪,⊆) and (⊆, ≪, ⊆), respectively, are founded, and conditions for a fuzzy soft set to be a fuzzy soft weak hyper BCK-ideal are considered. [ABSTRACT FROM AUTHOR]

Using the belongs to relation (∍) and quasi-coincident with relation (q) between fuzzy points and fuzzy sets, the notions of an (∊, &#8714)-fuzzy sub-hoop, an (∊, ∊v q)-fuzzy sub-hoop and a (q, ∊v q)-fuzzy sub-hoop are introduced, and several properties are investigated. Characterizations of an (∊, ∊)-fuzzy sub-hoop and an (∊,∊ vq)-fuzzy sub-hoop are displayed. Relations between an (∊,∊)-fuzzy sub-hoop, an (∊,∊ vq)-fuzzy sub-hoop and a (q,∊ vq)-fuzzy sub-hoop are discussed. Conditions for a fuzzy set to be a (q,∊ vq)-fuzzy sub-hoop are considered, and condition for an (∊,∊ vq)-fuzzy sub-hoop to be a (q,∊ vq)-fuzzy sub-hoop are provided. [ABSTRACT FROM AUTHOR]