Algebraic Structure Research Articles

A New Neutrosophic Algebraic Structure II

A New Neutrosophic Algebraic Structure II

ON A FAMILY OF DISCRETE ND LADDER-TYPE OPERATORS CONSTRUCTED IN TERMS OF THE HERMITIAN TOEPLITZ COMMUTATOR OPERATOR Z_N

A development of an algebraic system with N-dimensional ladder-type operators associated with the discrete Fourier transform is described, following an analogy with the canonical commutation relations of the continuous case. It is found that a Hermitian Toeplitz matrix Z_N, which plays the role of the identity, is sufficient to satisfy the Jacobi identity and, by solving some compatibility relations, a family of ladder operators with corresponding Hamiltonians can be constructed. The behaviour of the matrix Z_N for large N is elaborated. It is shown that this system can be also realized in terms of the Heun operator W, associated with the discrete Fourier transform, thus providing deeper insight on the underlying algebraic structure.

Complete Solutions for the Approximate Synthesis of Spherical Four-Bar Function Generators

Abstract Kinematic synthesis to meet an approximate motion specification naturally forms a constrained optimization problem. Instead of using local methods, we conduct global design searches by direct computation of all critical points. The idea is not new, but performed at scale is only possible through modern polynomial homotopy continuation solvers. Such a complete computation finds all minima, including the global, which allows for a full exploration of the design space, whereas local solvers can only find the minimum nearest to an initial guess. We form equality-constrained objective functions that pertain to the synthesis of spherical four-bar linkages for approximate function generation. We consider the general case where all mechanism dimensions may vary and a more specific case that enables the placement of ground pivots. The former optimization problem is shown to permit 268 sets of critical points, and the latter permits 61 sets. Critical points are classified as saddles or minima through a post-process eigenanalysis of the projected Hessian. The discretization points are contained within the coefficients of the stationarity polynomials, so the algebraic structure of the synthesis equations remains invariant to the number of points. The results of our computational work were used to design a mechanism that coordinates the folding wings. We also use this method to parameterize mechanism dimensions for a family of hyperbolic curves.

A Study of n-ary Groupoids Based on Hesitant Fuzzy Sets

Fuzzy set theory plays a vital role in solving many complicated problems dealing with uncertainty. An [Formula: see text]-ary algebraic system as a generalization of algebraic structures that allow for operations involving more than two elements. They provide a natural framework for representing and manipulating complex relationships and interactions among multiple elements, which is essential for solving many real-world problems. Many authors have studied fuzzy set theory over an [Formula: see text]-ary algebraic systems and have provided many fruitful results. Hesitant fuzzy set theory is a further extension of fuzzy set theory, which allows for the representation of uncertainty in decision making where decision makers may have multiple possible choices or may be uncertain about the degree of membership of an element in a given set. No authors have studied, by combining hesitant fuzzy set theory with an [Formula: see text]-ary algebraic systems. This motivates us to study hesitant fuzzy set over an [Formula: see text]-ary algebraic system. In this paper, we have applied hesitant fuzzy set theory in an [Formula: see text]-ary algebraic systems and introduced the notions of hesitant fuzzy subgroupoid. We provide the characterization of hesitant fuzzy [Formula: see text]-ary subgroupoid over an [Formula: see text]-ary groupoids and have studied their related properties. As an application of hesitant fuzzy set over an [Formula: see text]-ary groupoids, we have developed the concept of normal hesitant fuzzy subgroupoids over an [Formula: see text]-ary groupoids and have studied their various properties.

Relaxations and Exact Solutions to Quantum Max Cut via the Algebraic Structure of Swap Operators

Relaxations and Exact Solutions to Quantum Max Cut via the Algebraic Structure of Swap Operators

From the XXZ chain to the integrable Rydberg-blockade ladder via non-invertible duality defects

Strongly interacting models often possess “dualities” subtler than a one-to-one mapping of energy levels. The maps can be non-invertible, as apparent in the canonical example of Kramers and Wannier. We analyse an algebraic structure common to the XXZ spin chain and three other models: Rydberg-blockade bosons with one particle per square of a ladder, a three-state antiferromagnet, and two Ising chains coupled in a zigzag fashion. The structure yields non-invertible maps between the four models while also guaranteeing all are integrable. We construct these maps explicitly utilising topological defects coming from fusion categories and the lattice version of the orbifold construction, and use them to give explicit conformal-field-theory partition functions describing their critical regions. The Rydberg and Ising ladders also possess interesting non-invertible symmetries, with the spontaneous breaking of one in the former resulting in an unusual ground-state degeneracy.

Semigroups, keis and groups induced by knot diagrams: an experimental investigation with automated reasoning

AbstractWe apply automated reasoning to investigation of finite factors of algebraic structures associated with knot diagrams. We refute “double size” conjecture about cancellative semigroups and keis.

Borsik’s Properties of Topological Spaces and Their Applications

Let X be an uncountable Polish space. L̆ubica Holá showed recently that there are 2c\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2^{\\mathfrak {c}}$$\\end{document} quasi-continuous real valued functions defined on the uncountable Polish space X that are not Borel measurable. Inspired by Holá’s result, we are extending it in two directions. First, we prove that if X is an uncountable Polish space and Y is any Hausdorff space with |Y|≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|Y|\\ge 2$$\\end{document} then the family of all non-Borel measurable quasi-continuous functions has cardinality ≥2c\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ge 2^{{\\mathfrak {c}}}$$\\end{document}. Secondly, we show that the family of quasi-continuous non Borel functions from X to Y may contain big algebraic structures.

MULTISET TOPOLOGICAL SPACE AS GENERALIZATION OF THE CLASSICAL TOPOLOGICAL SPACE VIA SUPPORT SET

Multiset as a genelization of the classical set has triggered the definition and introduction of some algebraic structures in classical set theory under multiset context such as multiset topological spaces. In this paper we defined a root (support) set of a multiset topological space and established that multiset topological spaces are a generalizations classical topological spaces via their root sets respectively.

Matrices of the Zero Divisor Graphs of Classes of 3-Radical Zero Completely Primary Finite Rings

The study of finite completely primary rings through the zero divisor graphs, the unit groups and their associated matrices, and the automorphism groups have attracted much attention in the recent past. For the Galois ring R′ and the 2-radical zero finite rings, the mentioned algebraic structures are well understood. Studies on the 3-radical zero finite rings have also been done for the unit groups and the zero divisor graphs Γ(R). However, the characterization of the matrices associated with these graphs has not been exhausted. It is well known that proper understanding of the classification of zero divisor graphs with diameter 2 and girth 3 can provide insights into the structure of commutative rings and their zero divisors. In this study, we consider a class of 3-radical zero completely primary finite rings whose diameter and girth are 2 and 3 respectively. We enhance the understanding of the structure of such rings by investigating their Adjacency, Laplacian and Distance matrices.

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.

Nearness Cosets of the Nearness Groups

In 2019, Öztürk et al. [Gamma semigroups on weak nearness approximation spaces, Journal of the International Mathematical Virtual Institute 9 (2019) 53–72] defined the first algebraic structure on weak nearness approximation spaces which is the gamma semigroup. After this study, the view on the nearness of algebraic structures has completely changed. This view was first expressed in a previous study in 2021 [M. A. Öztürk, Nearness [Formula: see text]-algebras, Journal of Algebraic Hyperstructures and Logical Algebras 2 (2021) 73–84]. Our aim is to give the concept of modulo in the nearness group [Formula: see text] as in the nearness group [Formula: see text], where [Formula: see text] is the set of integers. Afterward, based on the concept of modulo in the nearness group [Formula: see text], the nearness cosets of the nearness group [Formula: see text] are constructed and its basic properties are examined. Lastly, it is seen that Lagrange theorem is not valid for the nearness subgroups as compared to the usual subgroups.

Complex Pythagorean Normal Interval-Valued Fuzzy Aggregation Operators for Solving Medical Diagnosis Problem

This paper presents a new methodology for solving multiple-attribute decision-making problems (MADMs) using a complex Pythagorean normal interval-valued fuzzy set (CPNIVFS), which is an extended concept of a complex Pythagorean fuzzy set. Four types of different aggregating operations (AOs), including CPNIVF weighted averaging (CPNIVFWA), CPNIVF weighted geometric (CPNIVFWG), generalized CPNIVFWA (CGPNIVFWA), and generalized CPNIVFWG (CGPNIVFWG), are discussed. The scoring function, accuracy function, and operational laws of the CPNIVFS are defined. Algebraic structures, such as associative, distributive, idempotent, bounded, commutativity, and monotonicity properties, are also shown to be satisfied by complex Pythagorean normal interval-valued fuzzy numbers. Furthermore, an algorithm is proposed to solve the MADM problems based on the defined AOs. The proposed approach is then used for a medical diagnosis problem about brain tumors because computer science and machine tool technology are among the most important components of brain tumor research. The five types of brain tumors diagnosed in these patients are gliomas, meningiomas, metastases, embryonal tumors, and ependymomas. Several types of treatments are available, which are often combined as part of an overall treatment plan. Brain tumors can be treated in various ways, including surgery, radiation therapy, chemotherapy, immunotherapy, and clinical trials. Based on the comparisons and options gathered, the most suitable treatment can be chosen. In this regard, it is evident that the value of the integer ⅁\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Game $$\\end{document} plays a significant role in determining the model. The candidate models under consideration can be validated by comparing them with the previously proposed ones. The proposed technique is compared with the existing method to demonstrate its superiority and validity, and the results conclude that the former is more reliable and effective than the latter. Finally, the criteria are evaluated by expert assessments to determine the most appropriate options.

Post-Lie algebra structures for perfect Lie algebras

We study the existence of post-Lie algebra structures on pairs of Lie algebras ( g , n ) , where one of the algebras is perfect non-semisimple, and the other one is abelian, nilpotent non-abelian, solvable non-nilpotent, simple, semisimple non-simple, reductive non-semisimple or complete non-perfect. We prove several nonexistence results, but also provide examples in some cases for the existence of a post-Lie algebra structure. Among other results we show that there is no post-Lie algebra structure on ( g , n ) , where g is perfect non-semisimple, and n is s l 3 ( C ) . We also show that there is no post-Lie algebra structure on ( g , n ) , where g is perfect and n is reductive with a 1-dimensional center.

Simplifying algebraic expressions with brackets: Insights into Grade 10 learners’ structure sense through a study of their errors

Developing structure sense is an important part of learning algebra. We investigated learners’ structure sense of algebraic expressions involving brackets. This led us to propose the constructs surface structure sense and systemic structure sense. Using a random sample of 58 Grade 10 learners scoring above 40% in a test, we coded incorrect responses for surface and systemic structure errors. The initial analysis revealed that the presence of more brackets supports surface structure sense. However, learners overgeneralised the presence of brackets to represent multiplication in situations involving subtraction. The arrangement of brackets also led to errors in the order of operations. Further analysis suggested that regular application of procedures on familiar algebraic structures may give the illusion of systemic structure sense. We recommend that the teaching of algebraic expressions must emphasise what the arrangement of an expression means before focusing on how to operate on the expression.Contribution: The research contributes to mathematics teaching by suggesting teaching strategies to improve on learners’ understanding of the role of brackets in algebraic expressions, by considering the arrangement of the components in structures.

SEMI GROUP IDENTITIES WITH APPLICATIONS TO SEMI GROUPS

This research paper explores the rich and intriguing world of semi-group identities, their properties, and their applications to various types of semi-groups. Semi-groups are algebraic structures that generalize the notion of groups, allowing for non-invertible elements. Despite their broader scope, semi-groups retain many important features found in group theory. This study investigates the identities that hold true in the context of semi-group algebra and sheds light on the underlying mathematical structures and relationships among these identities. By delving into specific applications, we illustrate the significance of these findings to various types of semi-groups, such as monoids, semigroups with zero, and cancellative semi-groups. Ultimately, our results not only deepen our understanding of the fundamental properties of semi-groups. But also provide valuable insights for researchers in the areas of algebraic structures, combinatorics, and theoretical computer science.

Wall-crossing effects on quiver BPS algebras

BPS states in supersymmetric theories can admit additional algebro-geometric structures in their spectra, described as quiver Yangian algebras. Equivariant fixed points on the quiver variety are interpreted as vectors populating a representation module, and matrix elements for the generators are then defined as Duistermaat-Heckman integrals in the vicinity of these points. The well-known wall-crossing phenomena are that the fixed point spectrum establishes a dependence on the stability (Fayet-Illiopolous) parameters ζ, jumping abruptly across the walls of marginal stability, which divide the ζ-space into a collection of stability chambers — “phases” of the theory. The standard construction of the quiver Yangian algebra relies heavily on the molten crystal model, valid in a sole cyclic chamber where all the ζ-parameters have the same sign. We propose to lift this restriction and investigate the effects of the wall-crossing phenomena on the quiver Yangian algebra and its representations — starting with the example of affine super-Yangian \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext{Y}}\\left({\\widehat{\\mathfrak{g}\\mathfrak{l}}}_{1\\left|1\\right.}\\right)$$\\end{document}. In addition to the molten crystal construction more general atomic structures appear, in other non-cyclic phases (chambers of the ζ-space). We call them glasses and also divide in a few different classes. For some of the new phases we manage to associate an algebraic structure again as a representation of the same affine Yangian \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext{Y}}\\left({\\widehat{\\mathfrak{g}\\mathfrak{l}}}_{1\\left|1\\right.}\\right)$$\\end{document}. This observation supports an earlier conjecture that the BPS algebraic structures can be considered as new wall-crossing invariants.

Classification of classical spin liquids: Detailed formalism and suite of examples

The hallmark of highly frustrated systems is the presence of many states close in energy to the ground state. Fluctuations between these states can preclude the emergence of any form of order and lead to the appearance of spin liquids. Even on the classical level, spin liquids are not all alike: they may have algebraic or exponential correlation decay, and various forms of long wavelength description, including vector or tensor gauge theories. Here, we introduce a classification scheme, allowing us to fit the diversity of classical spin liquids (CSLs) into a general framework as well as predict and construct new kinds. CSLs with either algebraic or exponential correlation-decay can be classified via the properties of the bottom flat band(s) in their soft-spin Hamiltonians. The classification of the former is based on the algebraic structures of gapless points in the spectra, which relate directly to the emergent generalized Gauss's laws that control the low-temperature physics. The second category of CSLs, meanwhile, are classified by the fragile topology of the gapped bottom band(s). Utilizing the classification scheme we construct new models realizing exotic CSLs, including one with anisotropic generalized Gauss's laws and charges with subdimensional mobility, one with a network of pinch-line singularities in its correlation functions and a series of fragile topological CSLs connected by zero-temperature transitions. Published by the American Physical Society 2024

Higher Dimensional Leibniz-Rinehart Algebras

In this article, we delve into the realm of higher dimensional Leibniz-Rinehart algebras, exploring the intricate structures of Leibniz algebroids and their applications. By generalizing the concept of Lie algebroids and incorporating a Leibniz rule for the anchor map, the study sheds light on the fundamental principles underlying connections and underscores their significance. Through a comprehensive analysis of Leibniz-Rinehart algebras, this study paves the way for advancements and applications, offering a deeper understanding of the intricate relationship between algebraic and geometric structures.

A DG-Enhancement of Dqc(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {D}}_{qc}(X)$$\\end{document} with Applications in Deformation Theory

It is well-known that DG-enhancements of the unbounded derived category Dqc(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {D}}_{qc}(X)$$\\end{document} of quasi-coherent sheaves on a scheme X are all equivalent to each other. Here we present an explicit model which leads to applications in deformation theory. In particular, we shall describe three models for derived endomorphisms of a quasi-coherent sheaf F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {F}$$\\end{document} on a finite-dimensional Noetherian separated scheme (even if F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {F}$$\\end{document} does not admit a locally free resolution). Moreover, these complexes are endowed with DG-Lie algebra structures, which we prove to control infinitesimal deformations of F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {F}$$\\end{document}.