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Symmetry properties are of vital importance for graphs. The famous Cayley graph is a good mathematical model as its high symmetry. The normality of the graph can well reflect the symmetry of the graph. In this paper, we characterize the normality of the direct product of Cayley graphs and give a sufficient and necessary condition for the direct product of two graphs to be a Cayley graph. Moreover, a sufficient and necessary condition for judging the normality of the direct product of two graphs is given. [ABSTRACT FROM AUTHOR]

In this research, we introduce the notion of an L -soft and investigate several universal properties of it. Then, the family of L -softs is established and denoted by Soft (). So, we tried to build an algebraic structure on Soft (). First, we showed that Soft () with sub- L operation is a poset. Also, we proved that Soft () with some operations made a complete distributive bounded lattice, bounded hoop, bounded L-algebra, CKL-algebra, BCK-algebra, commutative integral quantum B-algebra and Boolean algebra. Finally, we presented a congruence relation on Soft () and we investigated the quotient of it. [ABSTRACT FROM AUTHOR]

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]

Differential geometry of a kind of helical hypersurface family that depends on six parameters within the seven‐dimensional Euclidean space is explored. The curvatures of these hypersurfaces are determined, and their minimality is examined, along with illustrative examples being provided. Lastly, the Laplace–Beltrami operator for that kind of hypersurfaces is calculated. [ABSTRACT FROM AUTHOR]

The single‐valued neutrosophic set (SVNS) is a more common platform for representing the degree of truth, indeterminacy, and falsity memberships. Indeterminacy is an important aspect of SVNS, and the score function (SF) is important in ranking alternatives in decision‐making scenarios. When the score and accuracy values for different SVNNs were equal, existing SFs were unable to rank the alternatives. To address the limitations, this paper introduces the perfect score function (PSF) as a solution for ranking single‐valued neutrosophic numbers (SVNNs) without any errors and presents the concepts of strong and weak SVNSs. This article has emphasized the drawbacks of the existing SF in SVNNs. Additionally, we propose an optimized weighted geometric operator (OWGO) for SVNNs, and its properties such as idempotency, boundedness, monotonicity, and commutativity are explored. Furthermore, the decision‐making approach has been depicted based on the proposed score function and OWGO to select the finest faculty for a university. [ABSTRACT FROM AUTHOR]

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]

Spectral graph theory explores the relationship between the eigenvalues and the structural properties of graphs. This study focuses on the fundamental matrices associated with simple graphs, specifically the adjacency matrix. This work contributes to the spectral graph theory literature by introducing a new family of simple connected graphs and investigating their eigenvalue properties. We examine conditions under which these graphs satisfy specific spectral properties, such as the strong antireciprocal eigenvalue property. Additionally, we construct another family of graphs from this new family, proving that each graph within this derived family satisfies the strong antireciprocal eigenvalue property. Through these contributions, we aim to deepen the understanding of the spectral characteristics of graph families and their applications in theoretical and applied contexts. [ABSTRACT FROM AUTHOR]

The notion of θ-generalized int-soft subfields, θ-generalized int-soft algebras over θ-generalized int-soft subfields, and θ-generalized int-soft hypervector spaces are introduced, and their properties and characterizations are considered. In connection with linear transformations, θ-generalized int-soft hypervector spaces are discussed.

In a right regular L A -semigroup, the hybrid ideals, hybrid interior ideals, hybrid (1,2)-ideals, hybrid generalized bi-ideals, and hybrid quasi-ideals are all introduced in this work. A right regular L A -semigroup has also been described in terms of its hybrid right and hybrid left ideals. Moreover, we obtain some equivalent conditions in terms of hybrid two-sided ideal, hybrid bi- (respectively, generalized bi-)ideal, hybrid quasi-ideal, hybrid interior ideal and hybrid (1 , 2) -ideal over right regular L A -semigroups. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of (∈, ∈ ∨(κ⋇, q...))-anti-fuzzy b-ideal (AFBI) in BCI-algebra and explores its properties. It can be extended the notion of(∈ ∨(... ⋇, q...), ∈ ∨(... ⋇, q...))-AFBI is then presented and related properties are explored. We discuss the relations between (∈, ∈ ∨(... ⋇, q...))-AFBI and (∈ ∨(... ⋇, q...), ∈ ∨(... ⋇, q...))-AFBI. Furthermore, the concept of (∈, ∈ ∨(... ⋇, q...))-level subset applied to anti-fuzzy b-ideal. [ABSTRACT FROM AUTHOR]

One of the most critical concepts in the study of groups is the notion of nilpotency. Nilpotent groups arise in Galois theory, as well as in the classification of groups. In this paper, we apply the notion of nilpotency on B C K -algebras by using fuzzy ideals. First we define the concept of nilpotent fuzzy ideal in B C K -algebras and investigate their properties. Furthermore, we find some relation between fuzzy nilpotent ideals and quotient B C K -algebras. Specially, we construct a nilpotent B C K -algebra by using a fuzzy nilpotent ideal. Finally, we prove that any minimal ideals of a B C K -algebra X , is contained in the center of any fuzzy nilpotent ideals of X. [ABSTRACT FROM AUTHOR]

In this paper, the notion of closed cubic intuitionistic ideals, cubic intuitionistic p-ideals and cubic intuitionistic a-ideals in BCI-algebras are introduced, and several related properties are investigated. Relations between cubic intuitionistic subalgebras, closed cubic intuitionistic ideals, cubic intuitionistic q-ideals, cubic intuitionistic p-ideals and cubic intuitionistic a-ideals are discussed. Conditions for a cubic intuitionistic ideal to be a cubic intuitionistic p-ideal are provided. Characterizations of a cubic intuitionistic a-ideal are considered. The cubic intuitionistic extension property for a cubic intuitionistic a-ideal is established.

In this paper, we apply the m -polar cubic structures on BCI-algebras. We first present the ideas of m -polar cubic p -ideals, m -polar cubic q -ideals and m -polar cubic a -ideals in BCI-algebras. Then we prove that m -polar cubic p (q and a)-ideals are m -polar cubic ideals, but the converse assertion is not true, and an example is provided in this support. We define the conditions under which m -polar cubic ideals coincide with m -polar cubic p (q and a)-ideals. The associated properties of m -polar cubic p -ideals, m -polar cubic q -ideals, and m -polar cubic a -ideals are also examined. Furthermore, we consider m -polar cubic p (q and a)-ideals in terms of p (q and a)-ideals of BCI-algebras. [ABSTRACT FROM AUTHOR]

In this paper, we use IVI-octahedron ideals and IVI-octahedron bi-ideals to discuss some characterizations of special semigroups, e.g., an intra-regular semigroup, a semigroup which is semilattice of left [respectively, right] simple semigroups or simple left groups and a semigroup which is a semilattice of groups and a [respectively, left and right] simple semigroups. Finally, we define an IVIO-simple [IVIOL-simple and IVI) R-simple] semigroup and study some of its properties and we give a characterization of a semisimple semigroup by IVIOIs. [ABSTRACT FROM AUTHOR]

Our aim is to explore the idea of neutrosophic N−ideals in near-subtraction semigroups in this article and obtain some outcomes that are equivalent to them. We also illustrate the notion of a neutrosophic κ− intersection. Additionally, in a near-subtraction semigroup, we examine the term homomorphism of a neutrosophic κ− structure and establish some conclusions based on a homomorphic neutrosophic κ− structure preimage of a neutrosophic κ− left (respectively, right) ideal. [ABSTRACT FROM AUTHOR]

An AG-groupoid is the midway between commutative semigroup and groupoid. The core structure of Flock theory is an AG-groupoid, which focuses on motion replication and distance optimization and has numerous applications in physics and biology. Unfortunately, in many cases, modelling real-world problems in domains like computer science, operations research, artificial intelligence, control engineering, and robotics can be risky. Different theories, such as fuzzy sets, intuitionistic fuzzy sets, probability, soft sets, neutrosophic sets, and others, have been created to deal with similar situations. In this paper, We define the notions of neutrosophic κ-ideal structures in an AG-groupoid and investigate their properties. We also obtain equivalent assertion of neutrosophic κ-ideals and product of neutrosophic κ-structures in AG-groupoid. [ABSTRACT FROM AUTHOR]

In this paper, by considering the notion of fuzzy hyperoperation, we define the concept of fuzzy hyper B C K (B C I) -algebras as a generalization of B C K (B C I) -algebras and study related properties. Moreover, we introduce the notions of fuzzy hyper B C I -ideal and weak (strong) fuzzy hyper B C I -ideal, and then examine the relations among them. Finally, we investigate the relation between fuzzy hyper B C I -algebras with fuzzy hypersemigroup, fuzzy hypergroup and hyper B C I -algebras. [ABSTRACT FROM AUTHOR]

In this paper, the notions of ideals in NBCK-algebras are presented and some properties of such ideals are analyzed. Essentially, our approach is to define NBCK-algebras of all cosets by considering an operation on the set of all cosets defined via ideals. Afterwards, our aim is to study ideal theorems on NBCK-algebras. [ABSTRACT FROM AUTHOR]