Article
Mathematics
Oksana Bezushchak, Anatoliy Petravchuk, Efim Zelmanov
Summary: We prove analogs of A. Selberg's result for finitely generated subgroups of Aut(A) and of Engel's theorem for subalgebras of Der(A) for a finitely generated associative commutative algebra A over an associative commutative ring. We also prove an analog of the theorem of W. Burnside and I. Schur about local finiteness of torsion subgroups of Aut(A).
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
(2023)
Article
Physics, Multidisciplinary
Giovanni Landi, S. G. Rajeev
Summary: In this study, a Lie bi-algebra splitting of the algebra of smooth functions on the non-commutative torus T-theta(2) was constructed using the canonical trace. The Lie bi-algebra in the special case of rational parameter theta = M/N corresponds to unitary and upper triangular matrices. In the classical limit N -> infinity, the Lie bi-algebra has a remnant where the elements of U(N) tend to real functions and B(N) tends to a space of complex analytic functions, leading to an infinite dimensional Lie bi-algebra for smooth functions on the commutative torus.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2022)
Article
Computer Science, Theory & Methods
Lavinia Corina Ciungu
Summary: The aim of this paper is to define pseudo-valuations on quantum B-algebras and investigate their properties. Characterizations of pseudo-valuations are given and a relationship between positive strong pseudo-valuations and filters of a unital quantum B-algebra is shown. Pseudo-valuations are defined and characterized on specific cases of quantum B-algebras with product and pointed quantum B-algebras. Extension theorems are also proved for pseudo-valuations on good quantum B-algebras satisfying the Glivenko property, and potential generalizations of the Horn-Tarski extension theorem for pseudo-valuations on quantales are investigated.
FUZZY SETS AND SYSTEMS
(2023)
Article
Computer Science, Artificial Intelligence
Andrzej Walendziak
Summary: The study focuses on the commutative properties of various generalizations of pseudo-BCK algebras, providing an axiom system for commutative pseudo-aRML** algebras and proving their join-semilattice structure. It introduces concepts like commutative deductive systems, BB-deductive systems, and constructs quotient algebras, showing relationships between commutativity and specific conditions in these algebraic structures.
JOURNAL OF MULTIPLE-VALUED LOGIC AND SOFT COMPUTING
(2022)
Article
Mathematics
A. Alahmadi, H. Alsulami, S. K. Jain, E. Zelmanov
Summary: This paper investigates the universal enveloping algebra U(L) and the related Poisson algebra Pol(L) of finite dimensional Lie algebra L, and finds conditions for the finite generation of Lie algebras [U(L),U(L)] and [Pol(L),Pol(L)].
JOURNAL OF ALGEBRA
(2023)
Article
Mathematics, Applied
Louis Rowen, Yoav Segev
Summary: This paper deals with decomposition algebras which are non-commutative versions of primitive axial algebras of Jordan type (PAJs).
JOURNAL OF ALGEBRA AND ITS APPLICATIONS
(2023)
Article
Computer Science, Artificial Intelligence
Xiao Long Xin
Summary: This paper introduces a new structure called pseudo-EQ-algebras, which are a generalization of EQ-algebras. It discusses the differences between pseudo-EQ-algebras and pseudo-residuated lattices, as well as their related properties. The paper also explores the relationships between pseudo-EQ-algebras and other algebraic structures, as well as the connections to meet-semilattices groups.
Article
Mathematics
R. Garcia-Delgado
Summary: This work discusses the conditions for the existence of an invariant metric in the current Lie algebra $g \otimes \mathcal{S}$, where $g$ is a quadratic Lie algebra and $\mathcal{S}$ is an associative and commutative algebra with unit. The reciprocal case is also considered, where necessary and sufficient conditions are given for the existence of an invariant metric in $g$ if $g \otimes \mathcal{S}$ has one. In particular, it is shown that if $g$ is an indecomposable quadratic Lie algebra, $g \otimes \mathcal{S}$ has an invariant metric if and only if $\mathcal{S}$ has an invariant, symmetric, and non-degenerate bilinear form. Additionally, a theorem similar to double extension is proved for $g \otimes \mathcal{S}$, where $g$ is an indecomposable, nilpotent, and quadratic Lie algebra.
COMMUNICATIONS IN ALGEBRA
(2023)
Article
Mathematics
Ion Mihai, Radu-Ioan Mihai
Summary: The objective of this article is to establish the general Chen inequalities for statistical submanifolds in Hessian manifolds of constant Hessian curvature, extending previous results by different authors.
Article
Mathematics
Aliya Naaz Siddiqui, Ali Hussain Alkhaldi, Lamia Saeed Alqahtani
Summary: The geometry of Hessian manifolds is a fruitful branch of physics, statistics, Kaehlerian and affine differential geometry. This study mainly involves exploring inequalities for statistical submanifolds in Hessian manifolds of constant Hessian curvature.
Article
Mathematics, Applied
Pavel Osipov
Summary: This study discusses the properties of selfsimilar manifolds and selfsimilar Hessian manifolds, describing and characterizing the global and local cases.
JOURNAL OF GEOMETRY AND PHYSICS
(2022)
Article
Mathematics, Applied
Diogo Diniz, Dimas Jose Goncalves, Viviane Ribeiro Tomaz da Silva, Manuela Souza
Summary: In this paper, we classify the two-dimensional power-associative commutative algebras over an arbitrary field F of characteristic different from 2. As a consequence, we obtain a classification of two-dimensional Jordan algebras over F and prove the existence of a unique two-dimensional nonassociative Jordan algebra. We also generalize the construction of this algebra to produce a Jordan algebra D with an arbitrary dimension. Moreover, for infinite fields, we determine a finite basis for the polynomial identities of D, as well as of all associative Jordan algebras of dimension two. We also determine the codimension sequence of all these algebras, and if the field is of characteristic zero, we determine their cocharacter sequence. Thus, we conclude that the n-th codimension sequence of a two-dimensional Jordan algebra is bounded by n.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2023)
Article
Engineering, Electrical & Electronic
Alejandro Parada-Mayorga, Landon Butler, Alejandro Ribeiro
Summary: In this paper, we introduce and study the algebraic generalization of non commutative convolutional neural networks. By leveraging the theory of algebraic signal processing, we model convolutional non commutative architectures and derive stability bounds that extend those obtained for commutative convolutional neural networks. We show that non commutative convolutional architectures can be stable to deformations and analyze the trade-off between stability and selectivity, controlled by matrix polynomial functions.
IEEE TRANSACTIONS ON SIGNAL PROCESSING
(2023)
Article
Physics, Mathematical
Emily Cliff
Summary: In this paper, the authors review Borcherds's approach to vertex algebras and introduce new examples of his constructions. These examples are compared to vertex algebras, chiral algebras, and factorization algebras. The authors demonstrate that all vertex algebras, chiral algebras, or equivalently factorization algebras can be realized in these new categories VA(A, H, S).
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2022)
Article
Computer Science, Theory & Methods
Rui Paiva, Regivan Santiago, Benjamin Bedregal, Umberto Rivieccio
Summary: In this paper, a further generalization of BL-algebras without requiring associativity is proposed, based on a subclass of bivariate general overlap functions called inflationary. The differences between non-associative BL-algebras and the more specific class of inflationary BL-algebras are discussed, with a pictorial representation provided to summarize the relationship. Additionally, properties related to these algebras are proven, as well as a version of the Chinese Remainder Theorem under certain restrictions. The concepts of pseudo-automorphisms, automorphisms, and their effects on general overlap functions are utilized to obtain conjugated inflationary BL-algebras and to derive naBL-algebras from inflationary BL-algebras through automorphisms.
FUZZY SETS AND SYSTEMS
(2021)
