Research

13 pages, 286 KiB

Open AccessArticle

Characterization of Non-Linear Bi-Skew Jordan n-Derivations on Prime ∗-Algebras

by Asma Ali, Amal S. Alali and Mohd Tasleem

Axioms 2023, 12(8), 753; https://doi.org/10.3390/axioms12080753 - 30 Jul 2023

Viewed by 807

Let

A

be a prime *-algebra. A product defined as

U • V = U V^{*} + V U^{*}

for any

U, V \in A

, is called a bi-skew Jordan product. A map

ξ : A \to A

, [...] Read more.

Let

A

be a prime *-algebra. A product defined as

U • V = U V^{*} + V U^{*}

for any

U, V \in A

, is called a bi-skew Jordan product. A map

ξ : A \to A

, defined as

ξ (p_{n} (U_{1}, U_{2}, \dots, U_{n})) = \sum_{k = 1}^{n} p_{n} (U_{1}, U_{2}, . . ., U_{k - 1}, ξ (U_{k}), U_{k + 1}, \dots, U_{n})

for all

U_{1}, U_{2}, . . ., U_{n} \in A

, is called a non-linear bi-skew Jordan n-derivation. In this article, it is shown that

ξ

is an additive ∗-derivation. Full article

(This article belongs to the Special Issue Advances in Applied Algebra, Combinatorics and Computation)

12 pages, 275 KiB

Open AccessArticle

Fuzzy Hom–Lie Ideals of Hom–Lie Algebras

by Shadi Shaqaqha

Axioms 2023, 12(7), 630; https://doi.org/10.3390/axioms12070630 - 26 Jun 2023

Cited by 4 | Viewed by 921

Abstract

In the given study, we intended to gain familiarity with the idea of fuzzy Hom–Lie subalgebras (ideals) of Hom–Lie algebras. It primarily seeks to study a few of their properties. This research investigates the relationship between fuzzy Hom–Lie subalgebras (ideals) and Hom–Lie subalgebras [...] Read more.

In the given study, we intended to gain familiarity with the idea of fuzzy Hom–Lie subalgebras (ideals) of Hom–Lie algebras. It primarily seeks to study a few of their properties. This research investigates the relationship between fuzzy Hom–Lie subalgebras (ideals) and Hom–Lie subalgebras (ideals). Additionally, this study constructs new fuzzy Hom–Lie subalgebras based on the direct sum of a finite number of existing ones. Finally, the properties of fuzzy Hom–Lie subalgebras and fuzzy Hom–Lie ideals are examined in the context of the morphisms of Hom–Lie algebras. Full article

(This article belongs to the Special Issue Advances in Applied Algebra, Combinatorics and Computation)

11 pages, 275 KiB

Open AccessArticle

Applications of Fuzzy Semiprimary Ideals under Group Action

by Asma Ali, Amal S. Alali and Arshad Zishan

Axioms 2023, 12(6), 606; https://doi.org/10.3390/axioms12060606 - 19 Jun 2023

Cited by 1 | Viewed by 839

Abstract

Group actions are a valuable tool for investigating the symmetry and automorphism features of rings. The concept of fuzzy ideals in rings has been expanded with the introduction of fuzzy primary, weak primary, and semiprimary ideals. This paper explores the existence of fuzzy [...] Read more.

Group actions are a valuable tool for investigating the symmetry and automorphism features of rings. The concept of fuzzy ideals in rings has been expanded with the introduction of fuzzy primary, weak primary, and semiprimary ideals. This paper explores the existence of fuzzy ideals that are semiprimary but neither weak primary nor primary. Furthermore, it defines a group action on a fuzzy ideal and examines the properties of fuzzy ideals and their level cuts under this group action. In fact, it aims to investigate the relationship between fuzzy semiprimary ideals and the radical of fuzzy ideals under group action. Additionally, it includes the results related to the radical of fuzzy ideals and fuzzy

G

-semiprimary ideals. Moreover, the preservation of the image and inverse image of a fuzzy

G

-semiprimary ideal of a ring

R

under certain conditions is also studied. It delves into the algebraic nature of fuzzy ideals and the radical under

G

-homomorphism of fuzzy ideals. Full article

(This article belongs to the Special Issue Advances in Applied Algebra, Combinatorics and Computation)

8 pages, 274 KiB

Open AccessArticle

Minimality Conditions Equivalent to the Finitude of Fermat and Mersenne Primes

by Menachem Shlossberg

Axioms 2023, 12(6), 540; https://doi.org/10.3390/axioms12060540 - 31 May 2023

Viewed by 692

Abstract

The question is still open as to whether there exist infinitely many Fermat primes or infinitely many composite Fermat numbers. The same question concerning Mersenne numbers is also unanswered. Extending some recent results of Megrelishvili and the author, we characterize the Fermat primes [...] Read more.

The question is still open as to whether there exist infinitely many Fermat primes or infinitely many composite Fermat numbers. The same question concerning Mersenne numbers is also unanswered. Extending some recent results of Megrelishvili and the author, we characterize the Fermat primes and the Mersenne primes in terms of the topological minimality of some matrix groups. This is achieved by showing, among other things, that if

F

is a subfield of a local field of characteristic

\neq 2

, then the special upper triangular group

{ST}^{+} (n, F)

is minimal precisely when the special linear group

SL (n, F)

is. We provide criteria for the minimality (and total minimality) of

SL (n, F)

and

{ST}^{+} (n, F),

where

F

is a subfield of

C

. Let

F_{π}

and

F_{c}

be the set of Fermat primes and the set of composite Fermat numbers, respectively. As our main result, we prove that the following conditions are equivalent for

A \in {F_{π}, F_{c}}

:

A

is finite;

\prod_{F_{n} \in A} SL (F_{n} - 1, Q (i))

is minimal, where

Q (i)

is the Gaussian rational field; and

\prod_{F_{n} \in A} {ST}^{+} (F_{n} - 1, Q (i))

is minimal. Similarly, denote by

M_{π}

and

M_{c}

the set of Mersenne primes and the set of composite Mersenne numbers, respectively, and let

B \in {M_{π}, M_{c}} .

Then the following conditions are equivalent:

B

is finite;

\prod_{M_{p} \in B} SL (M_{p} + 1, Q (i))

is minimal; and

\prod_{M_{p} \in B} {ST}^{+} (M_{p} + 1, Q (i))

is minimal. Full article

(This article belongs to the Special Issue Advances in Applied Algebra, Combinatorics and Computation)

23 pages, 755 KiB

Open AccessArticle

Parametric Expansions of an Algebraic Variety near Its Singularities

by Alexander D. Bruno and Alijon A. Azimov

Axioms 2023, 12(5), 469; https://doi.org/10.3390/axioms12050469 - 13 May 2023

Cited by 1 | Viewed by 881

Abstract

Presently, there is a method based on Power Geometry that allows one to find asymptotic forms and asymptotic expansions of solutions to different kinds of non-linear equations near their singularities. The method contains three algorithms: (1) Reducing the equation to its normal form, [...] Read more.

Presently, there is a method based on Power Geometry that allows one to find asymptotic forms and asymptotic expansions of solutions to different kinds of non-linear equations near their singularities. The method contains three algorithms: (1) Reducing the equation to its normal form, (2) separating truncated equations, and (3) power transformations of coordinates. Here, we describe the method for the simplest case, a single algebraic equation, and apply it to an algebraic variety, as described by an algebraic equation of order 12 in three variables. The variety was considered in study of Einstein’s metrics and has several singular points and singular curves. Near some of them, we compute a local parametric expansion of the variety. Full article

(This article belongs to the Special Issue Advances in Applied Algebra, Combinatorics and Computation)

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13 pages, 313 KiB

Open AccessArticle

On Symbol-Pair Distance of a Class of Constacyclic Codes of Length 3p^s over

F_{p^{m}} + u F_{p^{m}}

by Hai Q. Dinh, Hiep L. Thi and Roengchai Tansuchat

Axioms 2023, 12(3), 254; https://doi.org/10.3390/axioms12030254 - 01 Mar 2023

Viewed by 883

Abstract

Let

p \neq 3

be any prime. In this paper, we compute symbol-pair distance of all

γ

-constacyclic codes of length

3 p^{s}

over the finite commutative chain ring

R = F_{p^{m}} + u F_{p^{m}}

, where

γ

[...] Read more.

Let

p \neq 3

be any prime. In this paper, we compute symbol-pair distance of all

γ

-constacyclic codes of length

3 p^{s}

over the finite commutative chain ring

R = F_{p^{m}} + u F_{p^{m}}

, where

γ

is a unit of

R

which is not a cube in

F_{p^{m}}

. We give the necessary and sufficient condition for a symbol-pair

γ

-constacyclic code to be an MDS symbol-pair code. Using that, we provide all MDS symbol-pair

γ

-constacyclic codes of length

3 p^{s}

over

R

. Some examples of the symbol-pair distance of

γ

-constacyclic codes of length

3 p^{s}

over

R

are provided. Full article

(This article belongs to the Special Issue Advances in Applied Algebra, Combinatorics and Computation)

12 pages, 264 KiB

Open AccessArticle

On the Left Properness of the Model Category of Permutative Categories

by Amit Sharma

Axioms 2023, 12(1), 87; https://doi.org/10.3390/axioms12010087 - 14 Jan 2023

Viewed by 886

Abstract

In this paper, we introduce a notion of free cofibrations of permutative categories. We show that each cofibration of permutative categories is a retract of a free cofibration. The main goal of this paper is to show that the natural model category of [...] Read more.

In this paper, we introduce a notion of free cofibrations of permutative categories. We show that each cofibration of permutative categories is a retract of a free cofibration. The main goal of this paper is to show that the natural model category of permutative categories is a left proper model category. Full article

(This article belongs to the Special Issue Advances in Applied Algebra, Combinatorics and Computation)

7 pages, 289 KiB

Open AccessArticle

On the Algebraic Independence of the Values of Functions Associated with Hypergeometric Functions

by Vasily Gorelov

Axioms 2023, 12(1), 36; https://doi.org/10.3390/axioms12010036 - 28 Dec 2022

Viewed by 986

Abstract

Functions that are integrals of products of generalized hypergeometric functions of some kind are considered. The conditions for the representability of these functions as a polynomial in hypergeometric ones are found. The necessary and sufficient conditions for the algebraic independence of such functions [...] Read more.

Functions that are integrals of products of generalized hypergeometric functions of some kind are considered. The conditions for the representability of these functions as a polynomial in hypergeometric ones are found. The necessary and sufficient conditions for the algebraic independence of such functions are established. Full article

(This article belongs to the Special Issue Advances in Applied Algebra, Combinatorics and Computation)

12 pages, 303 KiB

Open AccessArticle

Families of Ramanujan-Type Congruences Modulo 4 for the Number of Divisors

by Mircea Merca

Axioms 2022, 11(7), 342; https://doi.org/10.3390/axioms11070342 - 18 Jul 2022

Viewed by 1244

Abstract

In this paper, we explore Ramanujan-type congruences modulo 4 for the function

σ_{0} (n)

, counting the positive divisors of n. We consider relations of the form [...] Read more.

In this paper, we explore Ramanujan-type congruences modulo 4 for the function

σ_{0} (n)

, counting the positive divisors of n. We consider relations of the form

σ_{0} (8 (α n + β) + r) \equiv 0 (mod 4),

with

(α, β) \in N^{2}

and

r \in {1, 3, 5, 7}

. In this context, some conjectures are made and some Ramanujan-type congruences involving overpartitions are obtained. Full article

(This article belongs to the Special Issue Advances in Applied Algebra, Combinatorics and Computation)

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