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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]

In this paper, we introduced the concept of a soft hoop and we investigated some of their properties. Then, we established different types of intersections and unions of the family of soft hoops. We defined two operations ⊙ and → on the set of all soft hoops and we proved that with these operations, it is a hoop and also is a Heyting algebra. Finally we introduced a congruence relation on the set of all soft hoops and we investigated the quotient of it.

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]

After introducing the concept of single valued neutrosophic ideal in a pseudo MV-algebra, its properties are examined. The various conditions under which a single valued neutrosophic set can be a single valued neutrosophic ideal are examined. Characterizations of a single valued neutrosophic ideal of a pseudo MV-algebra are considered. [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]

We study hoops in order to give some new characterizations for regular and Boolean elements in hoops and we study the relationship between them. Specially, we prove that any bounded v̲-hoop is a Stone algebra if and only if MV -center set and Boolean elements set are equal. Then we define the concept of regular filter in hoops and v̲-hoops with RF-property and peruse some properties of them. In addition, we show that each Y-hoop with RF-property, is a Boolean algebra and any hoop A with RF-property such that B(A) = {0, 1}, is a local hoop. Finally, we prove that any hoop A has RF-property if and only if Spec(A) = Max(A) and if and only if A is a hyperarchimedean. [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]

In this paper, we defined the notions of (∈, ∈)-fuzzy implicative deductive systems and (∈, ∈ ∨ q)-fuzzy implicative deductive systems of hoops and studied some traits and tried to define some definitions that are equivalent to them. Thus by using the notion of (∈, ∈)-fuzzy deductive system of hoop, we defined a new congruence relation on hoop and show that the algebraic structure that is made by it is a Brouwerian semilattice, Heyting algebra and Wajesberg hoop. [ABSTRACT FROM AUTHOR]

In this paper, our aim was making a metric space on hoop algebras, because of that, we introduced the notion of valuation maps from F-quasi-valuation map based on hoops and related properties of them are investigated. By using these notions, we introduced a quasi-metric space. The continuity of operations of a hoop is studied with topology induced by a quasi-valuation. Also, we studied hoop homomorphism and investigated that under which condition this homomorphism is an F-quasi-valuation map. Moreover, we wanted to find a congruence relation on hoops in a new way and study about the quotient structure that is made by it. So, we defined a congruence relation by F-quasi-valuation map and proved that the quotient is a hoop. [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 this paper, we want to investigate some theorems about -cubic sets in [Sh. Rashid, M. Gulistan, Y. B. Jun, S. Khan and S. Kadry, -cubic sets and aggregation operators, J. Intell. Fuzzy Syst.37(4) (2019) 5009–5023] and while correcting these theorems, we get new results in this issue and at the end we present an application regarding decision making. First, by using some examples, we show that the conditions of some theorems in above-mentioned paper are not true in general and then we attempt to correct these theorems and proofs. Moreover, we discuss some new related results in -cubic sets. Finally, we discuss the score function and -cubic weighted average operator of -cubic sets for application in physical and different engineering regions. [ABSTRACT FROM AUTHOR]

Fuzzy graph theory is finding an increasing number of application in modeling real time systems where the level of information inherent in the system varies with different levels of precision. Special fuzzy graph can be obtained from two given fuzzy graphs using the operations beta products. In this paper, we introduce the notions of some kinds of beta product of two fuzzy graphs. The concept of strong, regular and complement of β i -product of two fuzzy graphs and relation between them are also obtained. At the end, an application with a cryptographic object is said to be using the β i -product of fuzzy graphs. [ABSTRACT FROM AUTHOR]

Recently, in Borzooei et al. [Inverse fuzzy graphs with applications, to appear in New Mathematics and Natural Computation], we defined the concept of inverse fuzzy graph as a generalization of graph, which is able to answer some problems that graph theory and fuzzy graph theory cannot explain. Now, in this paper, we define the notions of dominating set, domination number and different kinds of dominating sets like strong, weak, total, independent and inverse dominating sets with their domination numbers in inverse fuzzy graphs. Then, we investigate the relations among different kinds of dominating numbers. Finally, we give an application in decision making for service buildings. [ABSTRACT FROM AUTHOR]

The notion of DFS-filter (double-framed soft filter) in lattice implication algrebras is introduced and characteristics of DFS-filters are discussed and related result are investigated. We defined IDFS-filter (implicative double-framed soft filter) and extension property for IDFS-filters is established. We maked PIDFS-filter (positive implicative double-framed soft filter) and investigated its properties. Finally, relation between DFS-filter, IDFS-filter and PIDFS-filter discussed are. [ABSTRACT FROM AUTHOR]

The notion of &-equality algebras is introduced, and related properties are investigated. Using &-equality algebras, (commutative ordered) semigroups are induced. Conditions for an equality algebra to be an &-equality algebra and commutative equality algebra are provided. The concept of terminal section of an element is introduced, and several properties are studied. Using the notion of terminal section, conditions for an equality algebra to be a commutative equality algebra are considered. Given a subset of an equality algebra, the ◦-set and the ∗-set are introduced, and then several related properties are displayed. Conditions for the ◦-set (resp., ∗-set) to be deductive systems are stated. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of neutrosophic ℵ-bi-ideal for a semigroup. We infer different semigroups using neutrosophic ℵ -bi-ideal structures. Moreover, for regular semigroups, neutrosophic ℵ-product and intersection of neutrosophic ℵ-ideals are identical. [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]

The notion of neutrosophic quadruple a-ideal is introduced, and related properties are investigated. Relations between a neutrosophic quadruple p-ideal, a neutrosophic quadruple q-ideal, a neutrosophic quadruple a-ideal and a neutrosophic quadruple closed ideal are discussed. Conditions for the neutrosophic quadruple (A;B)-set Nq(A;B) to be a neutrosophic quadruple a-ideal are provided. [ABSTRACT FROM AUTHOR]

The notion of neutrosophic set theory is applied to lattice implication algebras, and the concept of neutrosophic LI-ideals and neutrosophic lattice ideals in a lattice implication algebra are introduced. Several properties are investigated. Relationships between a neutrosophic LI-ideal and a neutrosophic lattice ideal are established, and conditions for a neutrosophic lattice ideal to be a neutrosophic LI-ideal are provided. Characterizations of a neutrosophic LI-ideal are discussed. The properties of implication homomorphism of lattice implication algebras related to neutrosophic LI-ideals are studied. [ABSTRACT FROM AUTHOR]