Algebraic Identities Research Articles

Interval valued fuzzy ideals of near algebras

Interval valued fuzzy ideals of near algebras

Neutrosophic ℵ–Ideals (ℵ-Subalgebras) of Subtraction Algebra

The connection between neutrosophy and algebra has been of great interest with respect to many researchers. The objective of this paper is to provide a connection between neutrosophic ℵ−structures and subtraction algebras. In this regard, we introduce the concept of neutrosophic ℵ−ideals in subtraction algebra. Moreover, we study its properties and find a necessary and sufficient condition for a neutrosophic ℵ−structure to be a neutrosophic ℵ−ideal.

Validity of Closed Ideals in Algebras of Series of Square Analytic Functions

We show the validity of a complete description of closed ideals of the algebra is the algebra of series of analytic functions satisfying the Lipschitz condition of order αj2 obtained by.15

Codes in a Dihedral Group Algebra

In 1978, Robert McEliece constructed the first asymmetric code-based cryptosystem using noise-immune Goppa codes; no effective key attacks has been described for it yet. By now, quite a lot of code-based cryptosystems are known; however, their cryptographic security is inferior to that of the classical McEliece cryptosystem. In connection with the development of quantum computing, code-based cryptosystems are considered as an alternative to number theoretical ones; therefore, the problem of seeking promising classes of codes to construct new secure code-based cryptosystems is relevant. For this purpose, noncommutative codes can be used, that is, ideals in group algebras $${{\mathbb{F}}_{q}}G$$ over finite noncommutative groups $$G$$. The security of cryptosystems based on codes induced by subgroup codes has been studied earlier. The Artin–Wedderburn theorem, which proves the existence of an isomorphism of a group algebra to the direct sum of matrix algebras, is important for studying noncommutative codes. However, the particular form of terms and the construction of the isomorphism are not specified by this theorem; thus, for each group, there remains the problem of constructing the Wedderburn representation. The complete Wedderburn decomposition for the group algebra $${{\mathbb{F}}_{q}}{{D}_{{2n}}}$$ over the dihedral group $${{D}_{{2n}}}$$ has been obtained by F.E. Brochero Martinez in the case when the cardinality of the field and the order of the group are relatively prime numbers. Using these results, we study codes in the group algebra $${{\mathbb{F}}_{q}}{{D}_{{2n}}}$$ in this paper. The problem on the structure of all codes is solved, and the structure of codes induced by codes over cyclic subgroups of $${{D}_{{2n}}}$$ is described, which is of interest for cryptographic applications.

Algebras of block Toeplitz matrices with commuting entries

The maximal algebras of scalar Toeplitz matrices are known to be formed by generalized circulants. The identification of algebras consisting of block Toeplitz matrices is a harder problem that has received little attention up to now. We consider the case when the block entries of the matrices belong to a commutative algebra . After obtaining some general results, we classify all the maximal algebras for certain particular cases of .

Graded polynomial identities of triangular algebras

Let F be any field, G a finite abelian group and let A, B be F-algebras graded by subgroups of G. If M is a G-graded free (A,B)-bimodule, we describe the G-graded polynomial identities of the triangular algebra of M and, in case the field F has characteristic zero, we provide the description of its G-graded cocharacters by means of the graded cocharacters of A and B.

Jordan-Lie inner ideals of finite dimensional associative algebras

We study Jordan-Lie inner ideals of finite dimensional associative algebras and the corresponding Lie algebras and prove that they admit Levi decompositions. Moreover, we classify Jordan-Lie inner ideals satisfying a certain minimality condition and show that they are generated by pairs of idempotents.

A Note on Multiplicative (Generalized) (α, β)-Derivations in Prime Rings

Abstract Let R be a prime ring with center Z(R). A map G : R →R is called a multiplicative (generalized) (α, β)-derivation if G(xy)= G(x)α(y)+β(x)g(y) is fulfilled for all x; y ∈ R, where g : R → R is any map (not necessarily derivation) and α; β : R → R are automorphisms. Suppose that G and H are two multiplicative (generalized) (α, β)-derivations associated with the mappings g and h, respectively, on R and α, β are automorphisms of R. The main objective of the present paper is to investigate the following algebraic identities: (i) G(xy) + α(xy) = 0, (ii) G(xy) + α(yx) = 0, (iii) G(xy) + G(x)G(y) = 0, (iv) G(xy) = α(y) ○ H(x) and (v) G(xy) = [α(y), H(x)] for all x, y in an appropriate subset of R.

$L-$fuzzy semi-prime ideals in universal algebras

In this paper, we introduce the notion of $ L- $fuzzy semi-prime ideals and the radical of $ L- $fuzzy ideals in universal algebras and make a theoretical study on their basic properties.

Extension of Malcev Algebra and Applications to Gravity

AbstractWe investigate extensions of Malcev algebras and give an explicit example of extended algebras. We present a new algebraic identity, which can be regarded as a generalization of the Jacobi identity or the Malcev identity. As applications to gravity, we demonstrate that the extended algebra can be linked with general relativity.

Swapping Birth and Death: Symmetries and Transformations in Phylodynamic Models.

Stochastic birth–death models provide the foundation for studying and simulating evolutionary trees in phylodynamics. A curious feature of such models is that they exhibit fundamental symmetries when the birth and death rates are interchanged. In this article, we first provide intuitive reasons for these known transformational symmetries. We then show that these transformational symmetries (encoded in algebraic identities) are preserved even when individuals at the present are sampled with some probability. However, these extended symmetries require the death rate parameter to sometimes take a negative value. In the last part of this article, we describe the relevance of these transformations and their application to computational phylodynamics, particularly to maximum likelihood and Bayesian inference methods, as well as to model selection.

Voltage Differencing Buffered Amplifier based Voltage Mode Four Quadrant Analog Multiplier and its Applications

In this paper a voltage mode four quadrant analog multiplier (FQAM) using voltage differencing buffered amplifier (VDBA) based on quarter square algebraic identity is presented. In the proposed FQAM the passive resistor can be implemented using MOSFETs operating in saturationregion thereby making it suitable for integration. The effect of non idealities of VDBA has also been analyzed in this paper. Theoretical propositions are verified through SPICE simulations at 0.18μm CMOS technology node and the simulation results are found in close agreement with theoretical values. The supply voltage is taken as ± 1V and the value of the bias current is set to 40µA.The simulated total harmonic distortion (THD) is observed to be under 3% and the total power dissipation is found as 627µW. The workability of the proposed FQAM is also tested through two applications, namely, an amplitude modulator and a rectifier. The simulated results corroborate the theoretical propositions.

L-Fuzzy Prime Ideals in Universal Algebras

In this paper, the notions ofL-fuzzy prime ideals and maximalL-fuzzy ideals of universal algebras are introduced by applying the general theory of algebraic fuzzy systems.

Distinguishing simple algebras by means of polynomial identities

Our main goal is to extend one of classical Razmyslov’s Theorem saying that any two simple finite-dimensional \(\Omega \)-algebras over an algebraically closed field, satisfying the same polynomial identities, are isomorphic. We suggest a method that allows one to reduce problems about identities of algebras with additional structure to the identities of \(\Omega \)-algebras. For the convenience of the reader, we start with a full detailed proof of Razmyslov’s Theorem. Then we describe our method and its consequences for the identities of graded algebras, algebras with involution, and several others.

Combinatorics on Binary Words and Codimensions of Identities in Left Nilpotent Algebras

Numerical characteristics of polynomial identities of left nilpotent algebras are examined. Previously, we came up with a construction which, given an infinite binary word, allowed us to build a two-step left nilpotent algebra with specified properties of the codimension sequence. However, the class of the infinite words used was confined to periodic words and Sturm words. Here the previously proposed approach is generalized to a considerably more general case. It is proved that for any algebra constructed given a binary word with subexponential function of combinatorial complexity, there exists a PI-exponent, and its precise value is computed.

Quon Language: Surface Algebras and Fourier Duality

Quon language is a 3D picture language that we can apply to simulate mathematical concepts. We introduce the surface algebras as an extension of the notion of planar algebras to higher genus surface. We prove that there is a unique one-parameter extension. The 2D defects on the surfaces are quons, and surface tangles are transformations. We use quon language to simulate graphic states that appear in quantum information, and to simulate interesting quantities in modular tensor categories. This simulation relates the pictorial Fourier duality of surface tangles and the algebraic Fourier duality induced by the S matrix of the modular tensor category. The pictorial Fourier duality also coincides with the graphic duality on the sphere. For each pair of dual graphs, we obtain an algebraic identity related to the S matrix. These identities include well-known ones, such as the Verlinde formula; partially known ones, such as the 6j-symbol self-duality; and completely new ones.

On algebraic identities between solution matrices of Bessel's and Kummer's equations

On algebraic identities between solution matrices of Bessel's and Kummer's equations

Some commutativity theorems for rings with involution involving generalized derivations

Our purpose in this paper is to investigate commutativity of a ring with involution [Formula: see text] which admits a generalized derivation satisfying certain algebraic identities. Some well-known results characterizing commutativity of prime rings have been generalized. Moreover, we provide examples to show that the assumed restrictions cannot be relaxed.

$L$-fuzzy ideals in universal algebras

$L$-fuzzy ideals in universal algebras

Basic ideals in evolution algebras

With the aim of finding useful tools and invariants to classify finite dimensional evolution algebras, we introduce and study the notion of a basic ideal. Every n-dimensional perfect evolution algebra has a maximal basic ideal I which is unique except when the dimension of I is n-1. An application of our results leads to the description of the four dimensional perfect non-simple evolution algebras over a field with mild restrictions.