Class Of Algebraic Structures Research Articles

Some classes of semihypergroups of type U on the right

In this paper, we prove various inclusion relationships among different classes of algebraic structures and hyperstructures of type U on the right of finite size. In particular, we consider in detail the inclusion properties $$\mathcal{G}_n\subseteq \mathcal{PUR}_n\subseteq \mathcal{HUR}_n\subseteq \mathcal{SUR}_n$$ between the classes of groups $$\mathcal{G}_n$$, polygroups $$\mathcal{PUR}_n$$, hypergroups $$\mathcal{HUR}_n$$ and semihypergroups $$\mathcal{SUR}_n$$ of type U on the right of size n and provide conditions such that the equality holds. As a particular result, we prove that every polygroup of type U on the right is a group.

Hall's criterion for nilpotence in semi-abelian categories

A well-known theorem of P. Hall, usually called Hall's criterion for nilpotence, states: a group G is nilpotent whenever it has a normal subgroup N such that G/[N,N] and N are nilpotent. We widely generalize this result, replacing groups with objects in an abstract semi-abelian category satisfying suitable conditions. In particular, these conditions are satisfied in any algebraically coherent semi-abelian category, and hence in many categories of classical algebraic structures (including a few where Hall's theorem is already known). Note that the categories of crossed modules, crossed Lie algebras, n-cat groups, and cocommutative Hopf algebras over fields are also algebraically coherent. We give, however, a counter-example showing that Hall's theorem does not hold in the (semi-abelian) variety of non-associative rings.

Electrochemical Cells as Experimental Verifications of n-ary Hyperstructures

Algebraic hyperstructures represent a natural extension of classical algebraic structures and they have many applications in various sciences. The main purpose of this paper is to provide a new application of $n$-ary weak hyperstructures in Chemistry. More precisely, we present three different examples of ternary hyperstructures associated with electrochemical cells. In which we prove that our defined hyperstructures are ternary $H_{v}$-semigroups.

Image reconstruction based on circulant matrices

We propose a new method for image reconstruction based on circulant matrices. The novelty of this method is the image treatment using a simple and classical algebraic structure, the circulant matrix, which significantly reduces the computational effort, nevertheless providing reliable outputs. We compare the results with well established techniques such as the Principal Component Analysis (PCA) and the Discrete Fourier Transform (DFT), and the recently introduced Randomized Singular Value Decomposition (RSVD). We conclude that the quality is comparable whilst the computational time is considerably reduced.

Some Innovative Types of Fuzzy Ideals in AG-Groupoids

Abstract AG-groupoids (non-associative structure) are basic structures in Flocks theory. This theory mainly focuses on distance optimization, motion replication, and leadership maintenance with a wide range of applications in physics and biology. In this paper, we define some new types of fuzzy ideals of AG-groupoids called (α, β)-fuzzy bi-ideals, (α, β)-fuzzy interior ideals, (β̄, ᾱ)-fuzzy bi-ideals, and (β̄, ᾱ)-fuzzy interior ideals, where α, β∈{∈γ, qδ, ∈γ∨qδ, ∈γ∧qδ} and ᾱ, β̄∈{⋶γ, q̄δ, ⋶γ∨q̄δ, ⋶γ∧q̄δ}, with α≠∈γ∧qδ and β̄≠⋶γ∧q̄δ. An important milestone achieved by this paper is providing the connection between classical algebraic structures (ordinary bi-ideals, interior ideals) and new types of fuzzy algebraic structures [(∈γ, ∈γ∨qδ)-fuzzy bi-ideals, (∈γ, ∈γ∨qδ)-fuzzy interior ideals]. Special attention is given to (∈γ, ∈γ∨qδ)-fuzzy bi-ideals and (⋶γ, ⋶γ∨q̄δ) -fuzzy bi-ideals.

FUZZY RINGS AND ITS PROPERTIES

Abstract One of algebraic structure that involves a binary operation is a group that is defined an un empty set (classical) with an associative binary operation, it has identity elements and each element has an inverse. In the structure of the group known as the term subgroup, normal subgroup, subgroup and factor group homomorphism and its properties. Classical algebraic structure is developed to algebraic structure fuzzy by the researchers as an example semi group fuzzy and fuzzy group after fuzzy sets is introduced by L. A. Zadeh at 1965. It is inspired of writing about semi group fuzzy and group of fuzzy, a research on the algebraic structure of the ring is held with reviewing ring fuzzy, ideal ring fuzzy, homomorphism ring fuzzy and quotient ring fuzzy with its properties. The results of this study are obtained fuzzy properties of the ring, ring ideal properties fuzzy, properties of fuzzy ring homomorphism and properties of fuzzy quotient ring by utilizing a subset of a subset level and strong level as well as image and pre-image homomorphism fuzzy ring. Keywords: fuzzy ring, subset level, homomorphism fuzzy ring, fuzzy quotient ring

On the consistency problem for modular lattices and related structures

The consistency problem for a class of algebraic structures asks for an algorithm to decide, for any given conjunction of equations, whether it admits a non-trivial satisfying assignment within some member of the class. For the variety of all groups, this is the complement of the triviality problem, shown undecidable by by Adyan [Algorithmic unsolvability of problems of recognition of certain properties of groups. (Russian) Dokl. Akad. Nauk SSSR (N.S.) 103 (1955) 533–535] and Rabin [Recursive unsolvability of group theoretic problems, Ann. of Math. (2) 67 (1958) 172–194]. For the class of finite groups, it amounts to the triviality problem for profinite completions, shown undecidable by Bridson and Wilton [The triviality problem for profinite completions, Invent. Math. 202 (2015) 839–874]. We derive unsolvability of the consistency problem for the class of (finite) modular lattices and various subclasses; in particular, the class of all subspace lattices of finite-dimensional vector spaces over a fixed or arbitrary field of characteristic [Formula: see text] and expansions thereof, e.g. the class of subspace ortholattices of finite-dimensional Hilbert spaces. The lattice results are used to prove unsolvability of the consistency problem for (finite) rings with unit and (finite) representable relation algebras. These results in turn apply to equations between simple expressions in Grassmann–Cayley algebra and to functional and embedded multivalued dependencies in databases.

DIRAC COHOMOLOGY FOR DEGENERATE AFFINE HECKE-CLIFFORD ALGEBRAS

In this paper, we study the Dirac cohomology theory on a class of algebraic structures. The main examples of this algebraic structure are the degenerate affine Hecke-Clifford algebra of type An-1 by Nazarov and of classical types by Khongsap-Wang. The algebraic structure contains a remarkable subalgebra, which usually refers to Sergeev algebra for type An-1.We define an analogue of the Dirac operator for those algebraic structures. A main result is to relate the central characters of modules of those algebras with the central characters of modules of the Sergeev algebra via the Dirac cohomology. The action of the Dirac operator on certain modules is also computed. Results in this paper could be viewed as a projective version of the Dirac cohomology of the degenerate affine Hecke algebra.

Finitely generated equational classes

Classes of algebraic structures that are defined by equational laws are called varieties or equational classes. A variety is finitely generated if it is defined by the laws that hold in some fixed finite algebra. We show that every subvariety of a finitely generated congruence permutable variety is finitely generated; in fact, we prove the more general result that if a finitely generated variety has an edge term, then all its subvarieties are finitely generated as well. That is, finitely generated varieties with edge term are hereditarily finitely generated. This applies in particular to all varieties of groups, loops, quasigroups and their expansions (e.g., modules, rings, Lie algebras, …).

Relations on Groups, Polygroups and Hypergroups

In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. By using a certain type of equivalence relations, we can connect semihypergroups to semigroups and hypergroups (polygroups) to groups. These equivalence relations are called strongly regular relations. In this paper, we review some strongly regular relations on hyperstructures and we give some their applications.

Can Dirac quantization of constrained systems be fulfilled within the intrinsic geometry?

For particles constrained on a curved surface, how to perform quantization within Dirac’s canonical quantization scheme is a long-standing problem. On one hand, Dirac stressed that the Cartesian coordinate system has fundamental importance in passing from the classical Hamiltonian to its quantum mechanical form while preserving the classical algebraic structure between positions, momenta and Hamiltonian to the extent possible. On the other, on the curved surface, we have no exact Cartesian coordinate system within intrinsic geometry. These two facts imply that the three-dimensional Euclidean space in which the curved surface is embedded must be invoked otherwise no proper canonical quantization is attainable. In this paper, we take a minimum surface, helicoid, on which the motion is constrained, to explore whether the intrinsic geometry offers a proper framework in which the quantum theory can be established in a self-consistent way. Results show that not only an inconsistency within Dirac theory occurs, but also an incompatibility with Schrödinger theory happens. In contrast, in three-dimensional Euclidean space, the Dirac quantization turns out to be satisfactory all around, and the resultant geometric momentum and potential are then in agreement with those given by the Schrödinger theory.

The monads of classical algebra are seldom weakly cartesian

This paper begins a systematic study of weakly cartesian properties of monads that determine familiar varieties of universal algebras. While these properties clearly fail to hold for groups, rings, and many other related classical algebraic structures, their analysis becomes non-trivial in the case of semimodules over semirings, to which our main results are devoted. In particular necessary and sufficient conditions on a semiring S, under which the free semimodule monad has: (a) its underlying functor weakly cartesian, (b) its unit a weakly cartesian natural transformation, (c) its multiplication a weakly cartesian natural transformation, are obtained.

Exceptional Moufang quadrangles and structurable algebras

In 2000, J. Tits and R. Weiss classified all Moufang spherical buildings of rank two, also known as Moufang polygons. The hardest case in the classification consists of the Moufang quadrangles. They fall into different families, each of which can be described by an appropriate algebraic structure. For the exceptional quadrangles, this description is intricate and involves many different maps that are defined ad hoc and lack a proper explanation. In this paper, we relate these algebraic structures to two other classes of algebraic structures that had already been studied before, namely to Freudenthal triple systems and to structurable algebras. We show that these structures give new insight in the understanding of the corresponding Moufang quadrangles.

Inheritance examples of algebraic hyperstructures

Algebraic hyperstructures represent a natural extension of classical algebraic structures. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. Algebraic hyperstructure theory has many applications in other disciplines. The main purpose of this paper is to provide examples of hyperstructures associated with inheritance.

A physical example of algebraic hyperstructures: Leptons

Algebraic hyperstructures represent a natural extension of classical algebraic structures. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. Algebraic hyperstructure theory has a multiplicity of applications to other disciplines. The main purpose of this paper is to provide examples of hyperstructures associated with elementary particles in physical theory.

Free iterative and iteration K-semialgebras

We consider algebras of rational power series over a finite alphabet Σ with coefficients in a commutative semiring K and characterize them as the free algebras in various classes of algebraic structures.

Abstract Algebra in Statistics

In this note we show how some specific classes of algebraic structures (“planar nearrings”)give rise to efficient Balanced Incomplete Block Designs, which in turn can excellently beused in statistical experiments.

Dioïds and semirings: Links to fuzzy sets and other applications

Besides the classical algebraic structures of groups, rings and fields which have long been the almost exclusive reference concepts used in mathematical modelling, other algebraic structures such as dioïds and semirings have emerged in the last two or three decades in connection with modelling and solving a rich variety of non-classical problems, e.g. in Decision Analysis, Fuzzy Set Theory, Operations Research, Automatic Control and Mathematical Physics. The present paper aims at providing an overview of applications of dioïd and semiring structures, stressing links with Fuzzy Sets and emphasizing linear algebraic problems (solving linear systems, computing eigenvalues and eigenvectors), non-classical path-finding problems (using algebras of endomorphisms) and connections between dioïd structure and nonlinear analysis (in view of solving problems in Mathematical Physics).

Blocks of homogeneous effect algebras

Effect algebras, introduced by Foulis and Bennett in 1994, are partial algebras which generalise some well known classes of algebraic structures (for example orthomodular lattices, MV algebras, orthoalgebras et cetera). In the present paper, we introduce a new class of effect algebras, calledhomogeneous effect algebras. This class includes orthoalgebras, lattice ordered effect algebras and effect algebras satisfying the Riesz decomposition property. We prove that every homogeneous effect algebra is a union of its blocks, which we define as maximal sub-effect algebras satisfying the Riesz decomposition property. This generalizes a recent result by Riec˘anová, in which lattice ordered effect algebras were considered. Moreover, the notion of a block of a homogeneous effect algebra is a generalisation of the notion of a block of an orthoalgebra. We prove that the set of all sharp elements in a homogeneous effect algebraEforms an orthoalgebraEs. Every block ofEsis the centre of a block ofE. The set of all sharp elements in the compatibility centre ofEcoincides with the centre ofE. Finally, we present some examples of homogeneous effect algebras and we prove that for a Hilbert space ℍ with dim (ℍ) > 1, the standard effect algebra ℰ(ℍ) of all effects in ℰ is not homogeneous.

Relecture constructive de la théorie d'Artin-Schreier

We introduce the notion of “Dynamic Algebraic Structure” (DAS) inspired by Dynamic Evaluation (in computer algebra) and Model Theory. We show that this constructive notion allows a rereading of the Artin-Schreier-Robinson solution for the 17th Hilbert Problem. So, once we know how to reread the proofs, this kind of abstract theory contains an algorithm which computes the concrete result (here, the sum of squares required by Hilbert). Our method gives a constructive semantic for certain parts of abstract classical mathematics. The idea is the following: replace the classical algebraic structures “constructed” by Choice and Principle of Third Excluded Middle (TEM), by DAS and dynamic evaluations of these DAS. Then TEM is replaced by construction of branching in the trees of dynamic evaluation of the DAS. If Choice is used in the form of Godel completeness theorem, it is not really necessary to use it for obtaining concrete results: in DAS, Choice is simply replaced by … nothing!. This is because the classical proof is by contradiction: “if there were not a sum of squares then some formal theory would admit a pathological model”. The constructive reasoning is more direct: since the pathological theory proves 0 = 1 we know how to construct the sum of squares … and classical models have disappeared in the proof. They are replaced by dynamic evaluations of DAS. We think that we have given, for an academic example, a new method, realizing a kind of Hilbert Program for significant parts of classical algebra.