Class Of Algebraic Structures Research Articles

Modified double brackets and a conjecture of S. Arthamonov

Around 20 years ago, M. Van den Bergh introduced double Poisson brackets as operations on associative algebras inducing Poisson brackets under the representation functor. Weaker versions of these operations, called modified double Poisson brackets, were later introduced by S. Arthamonov in order to induce a Poisson bracket on moduli spaces of representations of the corresponding associative algebras. Moreover, he defined two operations that he conjectured to be modified double Poisson brackets. The first case of this conjecture was recently proved by M. Goncharov and V. Gubarev motivated by the theory of Rota-Baxter operators of nonzero weight. We settle the conjecture by realising the second case as part of a new family of modified double Poisson brackets. These are obtained from mixed double Poisson algebras, a new class of algebraic structures that are introduced and studied in the present work.

Soft Hyperstructures and Their Applications

The real world is full of uncertainties, which we require appropriate mathematical structures to model and handle them. One such structure is the soft set, which can be combined with various algebraic and topological structures. The paper discusses how soft sets and hyperstructures can be combined to create new soft structures that can handle uncertainty and imprecision in algebraic structures. Soft sets are mappings from parameters to subsets of a universal set, and they can model fuzzy concepts. Hyperstructures are generalizations of classical algebraic structures, where the composition of two elements is a set instead of a single element. They can capture complex and diverse phenomena that cannot be modeled by single-valued operations. The paper gives examples of soft structures, such as soft polygroups, soft topological polygroups, soft topological soft polygroups and their applications in mathematics and science. The paper aims to show the usefulness and versatility of soft structures in dealing with uncertainty and imprecision in algebraic structures. We will define the concept of soft topological soft polygroups and examine its characteristics.

Inversion operations in algebraic structures

We consider a wide series of classes of algorithmic complexity. We fix such a class and investigate the complexity of the inversion operation in classical algebraic structures, like groups or fields. In addition, we analyze the properties of quotient structures.

ALGEBRAIC SEMANTICS FOR RELATIVE TRUTH, AWARENESS, AND POSSIBILITY

AbstractThis paper puts forth a class of algebraic structures, relativized Boolean algebras (RBAs), that provide semantics for propositional logic in which truth/validity is only defined relative to a local domain. In particular, the join of an event and its complement need not be the top element. Nonetheless, behavior is locally governed by the laws of propositional logic. By further endowing these structures with operators—akin to the theory of modal Algebras—RBAs serve as models of modal logics in which truth is relative. In particular, modal RBAs provide semantics for various well-known awareness logics and an alternative view of possibility semantics.

Classes of Algebraic Structures

Classes of Algebraic Structures

Symbolic 4-Plithogenic Rings and 5-Plithogenic Rings

Symbolic n-plithogenic algebraic structures are considered as symmetric generalizations of classical algebraic structures because they have n + 1 symmetric components. This paper is dedicated to generalizing symbolic 3-plithogenic rings by defining symbolic 4-plithogenic rings and 5-plithogenic rings; these new classes of n-symbolic plithogenic algebraic structures will be defined for the first time, and their algebraic substructures will be studied. AH structures are considered to be a sign of the presence of symmetry within these types of ring, as they consist of several parts that are similar in structure and symmetrical, and when combined with each other, they have a broader structure resembling the classical consonant structure. Many related substructures will be presented such as 4-plithogenic/5-plithogenic AH-ideals, 4-plithogenic/5-plithogenic AH-homomorphisms, and 4-plithogenic AHS-isomorphisms will be discussed. We will show our results in terms of theorems, with many clear numerical examples that explain the novelty of this work.

On Novel Results about the Algebraic Properties of Symbolic 3-Plithogenic and 4-Plithogenic Real Square Matrices

Symbolic n-plithogenic sets are considered to be modern concepts that carry within their framework both an algebraic and logical structure. The concept of symbolic n-plithogenic algebraic rings is considered to be a novel generalization of classical algebraic rings with many symmetric properties. These structures can be written as linear combinations of many symmetric elements taken from other classical algebraic structures, where the square symbolic k-plithogenic real matrices are square matrices with real symbolic k-plithogenic entries. In this research, we will find easy-to-use algorithms for calculating the determinant of a symbolic 3-plithogenic/4-plithogenic matrix, and for finding its inverse based on its classical components, and even for diagonalizing matrices of these types. On the other hand, we will present a new algorithm for calculating the eigenvalues and eigenvectors associated with matrices of these types. Also, the exponent of symbolic 3-plithogenic and 4-plithogenic real matrices will be presented, with many examples to clarify the novelty of this work.

Punctured groups for exotic fusion systems

AbstractThe transporter systems of Oliver and Ventura and the localities of Chermak are classes of algebraic structures that model the ‐local structures of finite groups. Other than the transporter categories and localities of finite groups, important examples include centric, quasicentric, and subcentric linking systems for saturated fusion systems. These examples are, however, not defined in general on the full collection of subgroups of the Sylow group. We study here punctured groups, a short name for transporter systems or localities on the collection of nonidentity subgroups of a finite ‐group. As an application of the existence of a punctured group, we show that the subgroup homology decomposition on the centric collection is sharp for the fusion system. We also prove a Signalizer Functor Theorem for punctured groups and use it to show that the smallest Benson–Solomon exotic fusion system at the prime 2 has a punctured group, while the others do not. As for exotic fusion systems at odd primes , we survey several classes and find that in almost all cases, either the subcentric linking system is a punctured group for the system, or the system has no punctured group because the normalizer of some subgroup of order is exotic. Finally, we classify punctured groups restricting to the centric linking system for certain fusion systems on extraspecial ‐groups of order .

On Symbolic 2-Plithogenic Real Matrices and Their Algebraic Properties

Symbolic n-plithogenic sets came with many generalizations to classical algebraic structures, with many interesting properties and theorems, where the symbolic 2-plithogenic structures are very similar in their algebraic properties to refined neutrosophic algebraic structures. The main goal of this article is to study the algebraic properties of symbolic 2-plithogenic matrices such as the computing on symbolic 2-plithogenic determinants, symbolic 2-plithogenic special values, and symbolic 2-plithogenic representations by linear functions. In addition, many examples will be presented and discussed in terms of theorems to clarify the validity of the content of this paper.

Companions of fields of rational and real algebraic numbers

Companions of the field of rational numbers and a real-closed algebraic expansion of the field of rational numbers are studied. The description of existentially closed companions of a real-closed algebraic expansion of a field of rational numbers refers to the field of study of classical algebraic structures. The general theory of companions and existentially closed companions, built on the basis of Fraisse's classes in the works of A.T. Nurtazin, is included in the classical field of existentially closed theories in model theory. The basic concept of a companion: two models of the same signature are called companions if for any finite submodel of one of them, there is an isomorphic finite submodel in the other. This approach, applied to specific classical structures and their theories, provides new tools for the study of these objects. The study of the companion class of rational and algebraic real number fields reveals companion fields containing transcendental and possibly algebraic elements with special properties of polynomials defining these elements.

A New Neutrosophic Algebraic Structures

A new approach of neutrosophic algebraic structure is discussed in the work, which will open the door in front of researchers to new research about neutrosophic algebraic structure. In this work, we define new neutrosophic groupoid (semigroup, monoid) and new neutrosophic subgroupoid (subsemigroup, submonoid) in a new way which is more natural than the previous versions and we discuss some properties of these new neutrosophic concepts. Also, we discuss the relationship of new neutrosophic algebraic structures with other classical neutrosophic algebraic structure and prove some results. Finally, we introduced a new NeutroGroupoid and new NeutroSemiGroup.

Neutrosophic Submodule of Direct Sum M ⊕ N

The paper focuses on neutrosophic algebraic structures and operations applicability to the study of classical al-gebraic structures, particularly the R-module. The definition of neutrosophic submodules P and Q was further developed upon in this work in order to create neutrosophic submodules of P + Q. In this study, the neutrosophic submodule of the direct sum M ⊕ N is constructed, analyzed, and its associated results are examined. Additionally, several algebraic results of the neutrosophic submodule’s direct sum of a non-empty arbitrary family of submodules are examined.

On Neutrosophic Quadruple Groups

As generalizations and alternatives of classical algebraic structures there have been introduced in 2019 the NeutroAlgebraic structures (or NeutroAlgebras) and AntiAlgebraic structures (or AntiAlgebras). Unlike the classical algebraic structures, where all operations are well defined and all axioms are totally true, in NeutroAlgebras and AntiAlgebras, the operations may be partially well defined and the axioms partially true or, respectively, totally outer-defined and the axioms totally false. These NeutroAlgebras and AntiAlgebras form a new field of research, which is inspired from our real world. In this paper, we study neutrosophic quadruple algebraic structures and NeutroQuadrupleAlgebraicStructures. NeutroQuadrupleGroup is studied in particular and several examples are provided. It is shown that (NQ({mathbb {Z}}),div ) is a NeutroQuadrupleGroup. Substructures of NeutroQuadrupleGroups are also presented with examples.

About the Normal Projectivity and Injectivity of Krasner Hypermodules

Inspired by the concepts of projective and injective modules in classical algebraic structure theory, in this paper we initiate the study of the chains of hypermodules over a Krasner hyperring R, endowing first the set HomRn(M,N) of all normal homomorphisms between two R-hypermodules M and N with a structure of R-hypermodule. Then, our study focuses on the concepts of normal injectivity and projectivity of hypermodules over a Krasner hyperring R, characterizing them by the mean of chains of R-hypermodules.

Residuated Algebraic Structures in the Vicinity of Pre-rough Algebra and Decidability

Varieties of topological quasi-Boolean algebras in the vicinity of pre-rough algebras [28, 29] are expanded to residuated algebraic structures by introducing a new implication operation and its residual in these structures. Sequent calculi for some classes of residuated algebraic structures are established. These sequent calculi have the strong finite model property which yields the decidability of the word problem for corresponding classes of algebraic structures.

Indeterminacy in Neutrosophic Theories and their Applications

Indeterminacy makes the main distinction between fuzzy/intuitionistic fuzzy (and other extensions of fuzzy) set/logic vs. neutrosophic set/logic, and between classical probability and neutrosophic probability. Also, between classical statistics vs. neutrosophic and plithogenic statistics, between classical algebraic structures vs. neutrosophic algebraic structures, between crisp numbers vs. neutrosophic numbers. We present a broad definition of indeterminacy, various types of indeterminacies, and many practical applications.

Mini-Workshop: Algebraic Tools for Solving the Yang–Baxter Equation

The workshop was focused on three facets of the interplay between set-theoretic solutions to the Yang–Baxter equation and classical algebraic structures (groups, monoids, algebras, lattices, racks etc.): structures used to construct new solutions; structures as invariants of solutions; and YBE as a source of structures with interesting properties.

Classical algebraic structures in string theory effective actions

We study generic properties of string theory effective actions obtained by classically integrating out massive excitations from string field theories based on cyclic homotopy algebras of A∞ or L∞ type. We construct observables in the UV theory and we discuss their fate after integration-out. Furthermore, we discuss how to compose two subsequent integrations of degrees of freedom (horizontal composition) and how to integrate out degrees of freedom after deforming the UV theory with a new consistent interaction (vertical decomposition). We then apply our general results to the open bosonic string using Witten’s open string field theory. There we show how the horizontal composition can be used to systematically integrate out the Nakanishi-Lautrup field from the set of massless excitations, ending with a non-abelian A∞-gauge theory for just the open string gluon. Moreover we show how the vertical decomposition can be used to construct effective open-closed couplings by deforming Witten OSFT with a tadpole given by the Ellwood invariant. Also, we discuss how the effective theory controls the possibility of removing the tadpole in the microscopic theory, giving a new framework for studying D-brane deformations induced by changes in the closed string background.

On Proximity Spaces and Topological Hyper Nearrings

In 1934 the concept of algebraic hyperstructures was first introduced by a French mathematician, Marty. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the result of this composition is a set. In this paper, we prove some results in topological hyper nearring. Then we present a proximity relation on an arbitrary hyper nearring and show that every hyper nearring with a topology that is induced by this proximity is a topological hyper nearring. In the following, we prove that every topological hyper nearring can be a proximity space.

A Note on Neutrosophic Submodule of an R-module M

The paper focuses on the applications of neutrosophic set theory in the domain of classical algebraic structures, especially R-module. This study discusses some algebraic operations of neutrosophic sets of an R-moduleM, induced by the operations in M and demonstrates certain properties of the neutrosophic submodules of an R-module. The ideas of R module’s non-empty arbitrary family of neutrosophic submodules are characterized, and related outcomes are proved. The last section of this paper also derives a necessary and sufficient condition for a neutrosophic set of an R-module M.