10th Anniversary of Axioms: Algebra and Number Theory

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Algebra and Number Theory".

Deadline for manuscript submissions: closed (31 January 2023) | Viewed by 6443

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Prof. Dr. Oliver Stein

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Fakultät für Informatik und Mathematik, Ostbayerische Technische Hochschule Regensburg, Galgenbergstrasse 32, 93053 Regensburg, Germany
Interests: number theory; modular forms; automorphic forms; cryptography; mathematical cryptography; computation of elliptic curves and modular surfaces for cryptographic purposes
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Special Issue Information

Dear Colleagues,

Axioms launched its inaugural Issue in 2012, and is celebrating its 10th anniversary this year. We greatly appreciate the efforts and support from authors, reviewers, readers and editors over the last ten years. To mark this significant milestone and the achievements made throughout the years, we intend to publish a Special Issue entitled "10th Anniversary of Axioms: Algebra and Number Theory”.

This Special Issue aims to collect high-quality original research articles and reviews in advanced perspectives of mathematics. Contributions are welcomed which address topics including but not limited to number theory, algebraic and arithmetic geometry, group theory, and their subtopics. Studies of applications extended from this field are also encouraged, covering information science, theoretical physics, quantum chemistry, and so on. Many of the most interesting questions in each area remain open and are being actively worked on.

Prof. Dr. Oliver Stein
Guest Editor

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Published Papers (4 papers)

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Research

16 pages, 421 KiB

Open AccessArticle

On the Uniqueness of Lattice Characterization of Groups

by Jelena Jovanović, Branimir Šešelja and Andreja Tepavčević

Axioms 2023, 12(2), 125; https://doi.org/10.3390/axioms12020125 - 27 Jan 2023

Viewed by 968

Abstract

We analyze the problem of the uniqueness of characterization of groups by their weak congruence lattices. We discuss the possibility that the same algebraic lattice L acts as a weak congruence lattice of a group in more than one way, so that the corresponding diagonals are represented by different elements of L. If this is impossible, that is, if L can be interpreted as a weak congruence lattice of a group in a single way, we say that L is a sharp lattice. We prove that groups in many classes have a sharp weak congruence lattice. In particular, we analyze connections among isomorphisms of subgroup lattices of groups and isomorphisms of their weak congruence lattices. Summing up, we prove that there is a one-to-one correspondence between many known classes of groups and lattice-theoretic properties associated with each of these classes. Finally, an open problem is formulated related to the uniqueness of the element corresponding to the diagonal in the lattice of weak congruences of a group. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Algebra and Number Theory)

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16 pages, 861 KiB

Open AccessArticle

Centrally Extended Jordan (∗)-Derivations Centralizing Symmetric or Skew Elements

by Amal S. Alali, Hafedh M. Alnoghashi and Nadeem ur Rehman

Axioms 2023, 12(1), 86; https://doi.org/10.3390/axioms12010086 - 14 Jan 2023

Cited by 2 | Viewed by 1293

Abstract

Let A be a non-commutative prime ring with involution

*,

of characteristic

\neq 2 (and 3),

with Z as the center of A and

Π

a mapping

Π : A \to A

such that [...] Read more.

Let A be a non-commutative prime ring with involution

*,

of characteristic

\neq 2 (and 3),

with Z as the center of A and

Π

a mapping

Π : A \to A

such that

[Π (x), x] \in Z

for all (skew) symmetric elements

x \in A .

Π

is a non-zero CE-Jordan derivation of

A,

then A satisfies

s_{4},

the standard polynomial of degree

4 .

Π

is a non-zero CE-Jordan ∗-derivation of

A,

then A satisfies

s_{4}

Π (y) = λ (y - y^{*})

for all

y \in A,

and some

λ \in C,

the extended centroid of

A .

Furthermore, we give an example to demonstrate the importance of the restrictions put on the assumptions of our results. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Algebra and Number Theory)

9 pages, 600 KiB

Open AccessArticle

Combinatorial Interpretation of Numbers in the Generalized Padovan Sequence and Some of Its Extensions

by Renata Passos Machado Vieira, Francisco Regis Vieira Alves and Paula Maria Machado Cruz Catarino

Axioms 2022, 11(11), 598; https://doi.org/10.3390/axioms11110598 - 28 Oct 2022

Cited by 3 | Viewed by 1420

Abstract

There is ongoing research into combinatorial methods and approaches for linear and recurrent sequences. Using the notion of a board defined for the Fibonacci sequence, this work introduces the Padovan sequence combinatorial approach. Thus, mathematical theorems are introduced that refer to the study of the Padovan combinatorial model and some of its extensions, namely Tridovan, Tetradovan and its generalization (Z-dovan). Finally, we obtained a generalization of the Padovan combinatorial model, which was the main result of this research. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Algebra and Number Theory)

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Figure 1

7 pages, 248 KiB

Open AccessArticle

On Unique Factorization Modules: A Submodule Approach

by Sri Wahyuni, Hidetoshi Marubayashi, Iwan Ernanto and I Putu Yudi Prabhadika

Axioms 2022, 11(6), 288; https://doi.org/10.3390/axioms11060288 - 14 Jun 2022

Cited by 1 | Viewed by 1548

Abstract

Let M be a torsion-free module over an integral domain D. We define a concept of a unique factorization module in terms of v-submodules of M. If M is a unique factorization module (UFM), then D is a unique factorization domain. However, the converse situation is not necessarily to be held, and we give four different characterizations of unique factorization modules. Further, it is shown that the concept of the UFM is equivalent to Nicolas’s UFM, which is defined in terms of irreducible elements of D and M. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Algebra and Number Theory)

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