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The study of symmetry is a fascinating and unifying subject that connects various areas of mathematics in the twenty-first century. Algebraic structures offer a framework for comprehending the symmetries of geometric objects in pure mathematics. This paper introduces new concepts in algebraic structures, concentrating on semimodules over semirings and analysing the neutrosophic structure in this context. We explore the properties of neutrosophic subsemimodules and neutrosophic ideals after defining them. We discuss, utilizing neutrosophic products, the representations of neutrosophic ideals and subsemimodules, as well as the relationship between neutrosophic products and intersections. Finally, we derive equivalent criteria in terms of neutrosophic structures for a semiring to be fully idempotent. [ABSTRACT FROM AUTHOR]

The importance of symmetry in graph theory has always been significant, but in recent years, it has become much more so in a number of subfields, including but not limited to domination theory, topological indices, Gromov hyperbolic graphs, and the metric dimension of graphs. The purpose of this monograph is to initiate the idea of a multi polar q-rung orthopair fuzzy graphs (m- PqROPFG) as a fusion of multi polar fuzzy graphs and q-rung orthopair fuzzy graphs. Moreover, for a vertex of multi polar q-rung orthopair fuzzy graphs, the degree and total degree of the vertex are defined. Then, some product operations, inclusive of direct, Cartesian, semi strong, strong lexicographic products, and the union of multi polar q-rung orthopair fuzzy graphs (m- PqROPFG s) , are obtained. Also, at first we define some degree based fuzzy topological indices of m- PqROPFG. Then, we compute Zareb indices of the first and second kind, Randic indices, and harmonic index of a m- PqROPFG. [ABSTRACT FROM AUTHOR]

In this study, by generalizing the notion of fuzzy bi-ideals of ordered semirings, the notion of (∈ , ∈ ∨ (κ * , q κ)) -fuzzy bi-ideals is established. We prove that (∈ , ∈ ∨ (κ * , q κ)) -fuzzy bi-ideals are fuzzy bi-ideals but that the converse is not true, and an example is provided to support this proof. A condition is given under which fuzzy bi-ideals of ordered semirings coincide with (∈ , ∈ ∨ (κ * , q κ)) -fuzzy bi-ideals. An equivalent condition and certain correspondences between bi-ideals and (∈ , ∈ ∨ (κ * , q κ)) -fuzzy bi-ideals are presented. Moreover, the (κ * , κ) -lower part of (∈ , ∈ ∨ (κ * , q κ)) -fuzzy bi-ideals is described and depicted in terms of several classes of ordered semirings. Furthermore, it is shown that the ordered semiring is bi-simple if and only if it is (∈ , ∈ ∨ (κ * , q κ)) -fuzzy bi-simple. [ABSTRACT FROM AUTHOR]

In this paper, we study the properties of annihilator hyperideals in the class of strong bounded dual distributive meet-hyperlattice. We show that the set of all closed hyperideals forms a Boolean algebra. We introduce the concept of homomorphism, which preserves the annihilator hyperideal. Suitable conditions for preserving annihilator hyperideals are obtained. Representation and characterization theorems of annihilator hyperideals in sub-meet-hyperlattice and product meet-hyperlattice are proved. [ABSTRACT FROM AUTHOR]

We prove that in the ideal case, up to isomorphism, there are only one type of semigroups which are the union of two copies of the free monogenic semigroup. Similarly, there are only five types of semigroups which are the union of three copies of the free monogenic semigroup. And there are only two types of semigroups which are the union of two copies of the free semigroup in two generators. We provide finite presentations for each of these semigroups. [ABSTRACT FROM AUTHOR]