Search

In this paper, we will introduce and study the homological dimensions defined in the context of commutative rings with prime nilradical. So all rings considered in this paper are commutative with identity and with prime nilradical. We will introduce a new class of modules which are called ϕ -u-projective which generalizes the projectivity in the classical case and which is different from those introduced by the authors of [Y. Pu, M. Wang and W. Zhao, On nonnil-commutative diagrams and nonnil-projective modules, Commun. Algebra, doi:10.1080/00927872.2021.2021223; W. Zhao, On ϕ -exact sequence and ϕ -projective module, J. Korean Math. 58(6) (2021) 1513–1528]. Using the notion of ϕ -flatness introduced and studied by the authors of [G. H. Tang, F. G. Wang and W. Zhao, On ϕ -Von Neumann regular rings, J. Korean Math. Soc. 50(1) (2013) 219–229] and the nonnil-injectivity studied by the authors of [W. Qi and X. L. Zhang, Some Remarks on ϕ -Dedekind rings and ϕ -Prüfer rings, preprint (2022), arXiv:2103.08278v2 [math.AC]; X. Y. Yang, Generalized Noetherian Property of Rings and Modules (Northwest Normal University Library, Lanzhou, 2006); X. L. Zhang, Strongly ϕ -flat modules, strongly nonnil-injective modules and their homological dimensions, preprint (2022), https://arxiv.org/abs/2211.14681; X. L. Zhang and W. Zhao, On Nonnil-injective modules, J. Sichuan Normal Univ. 42(6) (2009) 808–815; W. Zhao, Homological theory over NP-rings and its applications (Sichuan Normal University, Chengdu, 2013)], we will introduce the ϕ -injective dimension, ϕ -projective dimension and ϕ -flat dimension for modules, and also the ϕ -(weak) global dimension of rings. Then, using these dimensions, we characterize several ϕ -rings (ϕ -Prüfer, ϕ -chained, ϕ -von Neumann, etc). Finally, we study the ϕ -(weak) global dimension of the trivial ring extensions defined by some conditions. [ABSTRACT FROM AUTHOR]

Quantum logic (QL) and fuzzy logic (FL) have been gaining attention nowadays due to its potential to be used in quantum computing and information theory. This paper aims to provide a comprehensive overview of QL models of fuzzy representations viz. fuzzy sets (FSs), interval valued fuzzy sets (IVFSs), intuitionistic fuzzy sets (IFSs) and interval valued intuitionistic fuzzy sets (IVIFSs). These QL models can be used to analyze the behavior of quantum logical systems in a fuzzy environment, which is particularly useful for dealing with uncertainty and imprecision in quantum environment. Furthermore, this paper explores the concept of effect algebras (EAs), which are algebraic structures that provide a natural framework for studying FL. Specifically, it is shown that the family of FS, IVFS, IFS and IVIFS can be organized into an EA if the Lukasiewicz operations are considered. This result is significant because EAs can be used to model a wide range of physical and mathematical systems, including quantum systems, and provide a useful tool for analyzing the properties of FL. [ABSTRACT FROM AUTHOR]

Over the years questions have arisen about T-ideals of (noncommutative) polynomials. But when evaluating a noncentral polynomial in subalgebras of matrices, one often has little control in determining the specific evaluations of the polynomial. One way of overcoming this difficulty in characteristic 0, is to reduce to multilinear polynomials and to utilize the representation theory of the symmetric group. But this technique is unavailable in characteristic p > 0. An alternative method, which succeeds, is the process of "hiking" a polynomial, in which one specializes its indeterminates in several stages, to obtain a polynomial in which Capelli polynomial is embedded, in order to get control on its evaluations. This method was utilized on homogeneous polynomials in the proof of Specht's conjecture for affine algebras over fields of positive characteristic. In this paper, we develop hiking further to nonhomogeneous polynomials, to apply to the "representability question." Kemer proved in 1988 that every affine relatively free PI algebra over an infinite field, is representable. In 2010, the first author of this paper proved more generally that every affine relatively free PI algebra over any commutative Noetherian unital ring is representable [A. Belov, Local finite basis property and local representability of varieties of associative rings, Izv. Russian Acad. Sci. (1) (2010) 3–134. English Translation Izv. Math. 74(1) (2010) 1–126]. We present a different, complete, proof, based on hiking nonhomogeneous polynomials, over finite fields. We then obtain the full result over a Noetherian commutative ring, using Noetherian induction on T-ideals. The bulk of the proof is for the case of a base field of positive characteristic. Here, whereas the usage of hiking is more direct than in proving Specht's conjecture, one must consider nonhomogeneous polynomials when the base ring is finite, which entails certain difficulties to be overcome. In Appendix A, we show how hiking can be adapted to prove the involutory versions, as well as various graded and nonassociative theorems. [ABSTRACT FROM AUTHOR]

In this paper, we first define twisted 풪-operator families on Leibniz algebras indexed by a semigroup Ω. Then we introduce and study NS-Leibniz family algebras as the underlying structures of twisted 풪-operator families. We show that an NS-Leibniz family algebra induces an ordinary NS-Leibniz algebra on the tensor product with the semigroup algebra. Finally, we investigate the cohomology of a twisted 풪-operator family. This cohomology can also be seen as the cohomology of a certain Ω-Leibniz algebra with coefficients in a suitable representation. As an application, we study the formal deformations of twisted 풪-operator families on Leibniz algebras. [ABSTRACT FROM AUTHOR]

In this paper, we first study clusters in type 픸̃ by collecting them into a finite number of infinite families given by Dehn twists of their corresponding triangulations, and show that these families are counted by the Catalan numbers. We also highlight the similarities and differences between the annuli diagrams used to study clusters and those used to study exceptional sets in type 픸̃. We then focus on exceptional collections (sets) of modules over path algebras of quivers by first showing that the notion of relative projectivity in exceptional sets is well defined. We finish by counting the number of exceptional sets of representations of type 픸 quivers with straight orientation and using this to count the number of families of exceptional sets of type 픸̃ with straight orientation. [ABSTRACT FROM AUTHOR]

In this paper, we establish the graded dimension formulas of finite quiver Hecke algebras using level-1 Fock space representations of quantum affine algebras of types A2n−1(2), Dn(1) and Bn(1). [ABSTRACT FROM AUTHOR]

In this paper, we introduce multi-layer quivers and show how to construct an (n + 1) -slice algebra of infinite type from an n -slice algebra of infinite type using the bound quivers. This leads to constructing (n + 1) -slice algebras of infinite type as matrix algebra and as tensor algebra of an n -slice algebra and equivalences of their module categories as the module categories of diagram of some quiver of type A ̃ n + 1 . [ABSTRACT FROM AUTHOR]

Co-Gorenstein algebras were introduced by Beligiannis in [A. Beligiannis, The homological theory of contravariantly finite subcategories: Auslander–Buchweitz contexts, Gorenstein categories and co-stabilization, Comm. Algebra28(10) (2000) 4547–4596]. In [S. Kvamme and R. Marczinzik, Co-Gorenstein algebras, Appl. Categorical Struct.27(3) (2019) 277–287], the authors propose the following conjecture (co-GC): if Ω n (m o d A) is extension closed for all n ≤ 1 , then A is right co-Gorenstein, and they prove that the generalized Nakayama conjecture implies the co-GC, also that the co-GC implies the Nakayama conjecture. In this paper, we characterize the subcategory Ω ∞ (m o d A) for algebras of Ω n -finite representation type. As a consequence, we characterize when a truncated path algebra is a co-Gorenstein algebra in terms of its associated quiver. We also study the behavior of Artin algebras of Ω ∞ -infinite representation type. Finally, an example of a non-Gorenstein algebra of Ω ∞ -infinite representation type and an example of a finite dimensional algebra with infinite ϕ -dimension are presented. [ABSTRACT FROM AUTHOR]

In this paper, we prove that an affine cellular algebra A is semisimple if and only if the scheme associated to A is reduced and 0-dimensional, and the bilinear forms with respect to all layers of A are invertible. Moreover, if the ground ring is a perfect field, then A is semisimple if and only if it is separable. We also give a sufficient condition for an affine cellular algebra being Jacobson semisimple. [ABSTRACT FROM AUTHOR]

A mathematical tool used to analyze signals in high-dimensional areas is the biquaternion ambiguity function. While typical ambiguity functions deal with time delay and Doppler frequency, biquaternion ambiguity functions extend the concept to rotations and scaling factors. Biquaternion ambiguity functions offer a more thorough representation of signal ambiguity in these high-dimensional spaces, which can help develop better signal processing algorithms for a range of uses. In this paper, we introduce the notion of biquaternion ambiguity function which is formed from conventional ambiguity function by invoking biquaternion algebra. We first establish the various properties, namely dilation, translation, shift, nonlinearity, boundedness, reconstruction formula, Moyal’s formula, and inversion formula. An interesting relationship between the biquaternion Fourier transform and the biquaternion ambiguity function is also established. Furthermore, we establish the various versions of uncertainty inequalities namely, Pitt’s inequality, sharp Hausdorff–Young inequality, sharp Local uncertainty principle and logarithmic uncertainty principle for biquaternion ambiguity function. Some potential applications are also presented at the end. [ABSTRACT FROM AUTHOR]

In [D. Rees, Hilbert functions and pseudo-rational local rings of dimension two, J. London Math. Soc. (2) 24 (1981) 467–479], Rees gave a characterization for the normal joint reduction number zero of two 픪-primary ideals in an analytically unramified Cohen–Macaulay local ring of dimension two. Rees’ result is a generalization of Zariski’s product theorem for complete ideals in a regular local ring of dimension two. The aim of this paper is to extend Rees’ theorem for the ordinary powers of 픪-primary ideals I and J in a Cohen–Macaulay local ring of dimension two. Following Rees’ approach, we define the modified Koszul homology modules Mr,s1(ak,bk) for a joint reduction (a,b) of I and J. Under the additional assumption that the associated graded rings of I and J have positive depth, we obtain a characterization of the joint reduction number zero of I and J in terms of the vanishing of the module M0,01(a,b), as well as in terms of the Hilbert coefficients and the bigraded Hilbert coefficients. More generally, we introduce the joint reduction lattice and study the vanishing of Mr,s1(a,b) for any r,s ≥ 0. This gives a characterization for a vector (r,s) to be in the joint reduction lattice of I and J. We also give a cohomological interpretation of these theorems by investigating the local cohomology modules of the bigraded extended Rees algebra. This gives another characterization for a vector (r,s) to be in the joint reduction lattice and also extends a recent result of Masuti and Verma in [Local cohomology of bigraded Rees algebras and normal Hilbert coefficients, J. Pure Appl. Algebra 218(5) (2014) 904–918] for ordinary powers of ideals. [ABSTRACT FROM AUTHOR]

In this paper, we present a method for finding the absorbing radical of a finite-dimensional evolution algebra. Such a method consists of finding the acyclic vertices of an oriented graph associated with the algebra. The set of generators associated with such vertices turn out to be the generators of the absorption radical. As an application we use the absorption radical to study the decomposability of some degenerate evolution algebras. [ABSTRACT FROM AUTHOR]

In this paper, the author gives a complete set of simple Yetter–Drinfeld modules over the Suzuki algebra A N 2 n + 1 μ λ and investigates the Nichols algebras over those irreducible Yetter–Drinfeld modules. The finite-dimensional Nichols algebras of diagonal type are of Cartan type A 1 , A 1 × A 1 , A 2 , Super type A 2 (q ; 2) and the Nichols algebra (8). And the involved finite-dimensional Nichols algebras of non-diagonal type are 1 2 , 4 m and m 2 -dimensional. The left three unsolved cases are set as open problems. In particular, all finite-dimensional Nichols algebras are given for simple Yetter–Dinfeld modules over  A 1 3 + λ . [ABSTRACT FROM AUTHOR]

For a rooted cluster algebra (Q) over a valued quiver Q , a symmetric cluster variable is any cluster variable belonging to a cluster associated with a quiver σ (Q) , for some permutation σ. The subalgebra of (Q) generated by all symmetric cluster variables, is called the symmetric mutation subalgebra and is denoted by ℬ (Q). In this paper, we identify the class of cluster algebras that satisfy ℬ (Q) = (Q) , which contains almost every quiver of finite mutation type. In the process of proving the main result, we provide a classification of quivers mutations classes that relates their maximum weights to the shapes of the initial quivers. Furthermore, some properties of symmetric mutation subalgebras are given. [ABSTRACT FROM AUTHOR]

In this paper, we study Whittaker modules over the N = 2 Neveu–Schwarz algebra 픤. We first classify all simple finite-dimensional modules over the positive part of 픤. Motivated by this classification, we define Whittaker modules and investigate their simplicity. Finally, we also study the simplicity of the quotient modules of the universal Whittaker module if it is not simple. [ABSTRACT FROM AUTHOR]

In this paper, we study automorphisms of finitely generated free metabelian Novikov algebras and show that every tame automorphism of a two-generated free right nilpotent Novikov algebra of index 3 is simple reducible. We offer a method on recognizing automorphisms of finitely generated free metabelian Novikov algebras by using the theory of Gröbner–Shirshov basis. [ABSTRACT FROM AUTHOR]

For the purposes of our analysis, we shall work with the Steenrod algebra 풜 over the field of two elements, 픽2. Our paper focuses on expanding the outcomes from our preceding study, which serves as Part I. Precisely, we will initially compute the dimension of the “cohit” space 픽2 ⊗풜Ps, where Ps is defined as the graded polynomial ring 픽2[x1,x2,…,xs :,deg(xi) = 1,i = 1, 2,…,s], for the specific case of s = 5. This computation will be carried out explicitly for the generic degree 21 ⋅ 2t − 5, where t is a positive integer. We have developed a novel algorithm implemented in SageMath to compute the case where t = 2. We subsequently study the dimension of 픽2 ⊗풜Ps in degrees s + 5 for 8 ≤ s ≤ 9. One of our findings is the correction of some erroneous results in the previous work by Moetele and Mothebe. We, additionally, give an explicit formula for the dimension of 픽2 ⊗풜Ps in degree fourteen for all s > 0 and in degree fifteen for all s > 0,s≠10. We also describe a computational method implemented in SageMath to explicitly account for the number of spikes of degree 2s−1 − s in the 풜-module Ps, applicable for all positive integers s > 1. In application, we examine W. Singer’s conjecture on the algebraic transfer of rank 5 in degrees 21 ⋅ 2t − 5 for all t ≥ 0 and of ranks s > 0 in internal degrees d ranging from 13 to 15. Considering higher homological degrees, we analyze the behavior of the algebraic transfer of rank 7 in the generic degrees 23 ⋅ 2t − 7 for t = 0 and ℓ ⋅ 2t − 7 for ℓ ∈{9, 16},t ≤ 3. We then demonstrate that the (in)decomposable elements Pc0 ∈Ext풜7,23⋅20(픽2, 픽2), k0 = k ∈Ext풜7,9⋅22(픽2, 픽2), h6D2 ∈Ext풜7,27(픽2, 픽2), Q2(0) ∈Ext풜7,26(픽2, 픽2) and Q2(1) ∈Ext풜7,27(픽2, 픽2) are not in the image of the Singer transfer. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a spinor algebra of k-balancing numbers referred to as the k-balancing and k-Lucas-balancing spinors. First, we give k-balancing quaternions and their some algebraic properties. Then we introduce a spinor family of k-balancing numbers by defining a linear and injective correspondence between k-balancing and k-Lucas-balancing quaternions to spinors. Here, we give various algebraic properties of this spinor such as the Binet form, Catalan’s identity, d’Ocagne identity, generating and exponential generating functions. Moreover, we obtain various partial sum formulae for these sequences in closed form and also give matrix representations. [ABSTRACT FROM AUTHOR]

For a curve in positive characteristic, the Cartier operator acts on the vector space of its regular differentials. The a-number is defined to be the dimension of the kernel of the Cartier operator. In [a-numbers of curves in Artin–Schreier covers, Algebra Number Theory 14(3) (2020) 587–641], Booher and Cais use a sheaf-theoretic approach to give bounds on the a-numbers of Artin–Schreier covers. In this paper, I generalize that approach to arbitrary powers of the Cartier operator, yielding bounds for the dimension of the kernel. These bounds give new restrictions on the Ekedahl-Oort type of Artin–Schreier covers. [ABSTRACT FROM AUTHOR]

In this paper, we study the powers of the generalized binomial edge ideal K m , P n of a path graph P n . We explicitly compute their regularities and determine the limit of their depths. We also show that these ordinary powers coincide with their symbolic powers. Additionally, we study the Rees algebra and the special fiber ring of K m , P n utilizing Sagbi basis theory. In particular, we obtain precise formulas for the regularity of these blowup algebras. [ABSTRACT FROM AUTHOR]

Polynomial completeness results aim at characterizing those functions that are induced by polynomials. Each polynomial function is congruence preserving, but the opposite need not be true. A finite algebraic structure A is called strictly 1-affine complete if every unary partial function from a subset of A to A that preserves the congruences of A can be interpolated by a polynomial function of A. The problem of characterizing strictly 1-affine complete finite Mal'cev algebras is still open. In this paper, we extend the characterization by Aichinger and Idziak of strictly 1-affine complete expanded groups to finite congruence regular Mal'cev algebras. [ABSTRACT FROM AUTHOR]

In this paper, some properties of vertex algebras acted by a Hopf algebra will be given. We prove that V # H is an -local vertex algebra, where V is an H -module vertex algebra. In addition, we extend the -locality to the quantum vertex algebras. [ABSTRACT FROM AUTHOR]

This paper is a contribution to the globalization problem for partial group actions on non-associative algebras. We principally focus on partial group actions on Lie algebras, Jordan algebras and Malcev algebras. We give sufficient conditions for the existence and uniqueness of a globalization for partial group actions on the algebras already mentioned. As an application of this result, we show that in characteristic zero every partial group action on a semisimple Malcev algebra admits a globalization, unique up to isomorphism. We give a criterion for the existence and uniqueness of a globalization for a partial group action on a unital Jordan algebra in characteristic different from two, and on a sympathetic Lie algebra (a perfect Lie algebra without center and outer derivations). [ABSTRACT FROM AUTHOR]

In this paper, we describe the so-called homogeneous involution on finite-dimensional graded-division algebra over an algebraically closed field. We also compute their graded polynomial identities with involution. As pointed out by Fonseca and Mello, a homogeneous involution naturally appears when dealing with graded polynomial identities and a compatible involution. [ABSTRACT FROM AUTHOR]

In this paper, we re-establish a bialgebraic theory on a Hom–Leibniz algebra by a new dual representation, which weakens the involutive condition on the structure map in [S. Guo, S. X. Wang and X. H. Zhang, The Classical Hom–Leibniz Yang–Baxter Equation and Hom–Leibniz Bialgebras, Mathematics 10 (2022) 1920; I. Laraiedh and S. Silvestrov, Hom–Leibniz bialgebras and BiHom–Leibniz dendriform algebras, Afr. Mat. 34 (2023), Article ID: 28, 45 pp]. We also give the classification of triangular Hom–Leibniz bialgebras on two-dimensional Hom–Leibniz algebras induced by “Yau twist principle”. [ABSTRACT FROM AUTHOR]

The n-th Schrödinger algebra 픰픠픥n := 픰픩2 ⋉ 픥n is the semidirect product of the Lie algebra 픰픩2 with the n-th Heisenberg Lie algebra 픥n. In this paper, we will show that 픰픠픥n has neither nontrivial 1 2-derivations nor nontrivial transposed Poisson algebra structures, and doesn’t have nonzero 1 2-biderivations. In addition, all δ-derivations and δ-biderivations on 픰픠픥n are all described. [ABSTRACT FROM AUTHOR]

In the this paper, as a generalization of the classical periodic rings, we explore those rings whose elements are additively generated by two (or more) periodic elements by calling them additively periodic. We prove that, in some major cases, additively periodic rings remain periodic too. This includes, for instance, algebraic algebras, group rings, and matrix rings over commutative rings. Moreover, we also obtain some independent results for the new class of rings. For example, the triangular matrix rings retain that property. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce a weighted derivation on algebras over an operad 풫, and prove that for the free 풫-algebra, its weighted derivation is determined by the restriction on the generators. As applications, we propose the concept of weighted differential (q-tri)dendriform algebras and study some basic properties of them. Then Novikov-(tri)dendriform algebras are initiated, which can be induced from differential (q-tri)dendriform of weight zero. Finally, the corresponding free objects are constructed, in both the commutative and noncommutative contexts. [ABSTRACT FROM AUTHOR]

Monounary algebras are the most simple type of an algebraic structure. Oriented graphs with one outgoing arrow from every vertex represent them. The aim of this paper is to point out the interdisciplinary relationships concerning this structure. Bernoulli shift is a paradigmatic mapping in dynamical systems. It is also called dyadic, bit shift, doubling or sawtooth. We offer a look at the properties of this mapping via monounary algebras. [ABSTRACT FROM AUTHOR]

In this paper, we shall study three types of Galois extensions B which are Azumaya algebras over its center C. (1) 1 , the class of Azumaya–Galois extensions; (2) 2 , the class of Galois extensions B of B G with Galois group G such that B G is a separable C G -algebra; and (3) 3 , the class of Galois extensions which are Azumaya algebras over its center C. Clearly, 1 ⊆ 2 ⊆ 3 . Examples are given to show the proper inclusion relationship 1 ⊂ 2 ⊂ 3 , and equivalent conditions are given under which a B ∈ 3 is in 1 and a B ∈ 2 is in 1 . [ABSTRACT FROM AUTHOR]

Assume that is a unital algebra over a commutative unital ring ℛ and is an -bimodule. A trivial extension algebra ⋉ is defined as an ℛ -algebra with usual operations of ℛ -module × and the multiplication defined by (u 1 , s 1) (u 2 , s 2) = (u 1 u 2 , u 1 s 2 + s 1 u 2) for all u 1 , u 2 ∈ , s 1 , s 2 ∈. In this paper, we prove that under certain conditions every Jordan n -derivation Δ on ⋉ can be expressed as Δ = d + δ , where d is a derivation and δ is both a singular Jordan derivation and an antiderivation. As applications, we characterize Jordan n -derivations on triangular algebras and generalized matrix algebras. [ABSTRACT FROM AUTHOR]

Let R be a commutative integral domain with unity that contains ℚ. In this paper we describe the R-automorphism group of R-algebras of the form R[x,y,z]/(c(x)z − q(x,y)), where c(x) ∈ R[x] is quasi-monic and q(x,y) ∈ R[x,y] is quasi-monic of degree at least two with respect to y. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a condition on a p-subgroup of a finite group that is more general than the concept of ICΦ-subgroups, and obtain a sufficient and necessary condition for p-nilpotence of a finite group. We not only simplify the proofs of some main theorems of Gao and Li [Acta Math. Hungar. 163(1) (2021) 29–36], Kaspczyk [Acta Math. Hungar. 165(2) (2021) 355–359], and Kaspczyk [J. Algebra Appl. 23(1) (2024) 2450016], but also generalize these results. [ABSTRACT FROM AUTHOR]

A set of position-dependent noncommutative algebra in two dimension (2D) that describes the space near the Planck scale had been introduced [J. Phys. A: Math. Theor. 53 (2020) 115303]. This algebra predicted the existence of maximal length of graviton measurable at low energy. From this algebra, we deduce in the present paper, a new noncommutative algebra that is compatible with the deformed algebra proposed by Costa Filho et al. [Phys. Rev. A84 (2011) 050102] to describe the Position-Dependent Mass (PDM) system in 1D. To this aim, we derive from the momentum operators, the Schrödinger-like equation which describes PDM system in null quantum well potential. The spectrum of this system is asymetrically deformed and exhibits different behaviours from the one obtained by Costa Filho et al. Thus, we observe that by increasing the PDM, the energy decreases and this decrease is more pronounced as the quantum number increases. Finally, we evaluate the thermodynamic quantities within the canonical ensemble and we show that these results are consistent with the ones recently obtained by Bensalem and Bouaziz [Phys. A Stat. Mech. Appl.523 (2019) 583–592]. [ABSTRACT FROM AUTHOR]

In this paper, we construct the super Virasoro algebra with an arbitrary conformal dimension Δ from the generalized ℛ (p , q) -deformed quantum algebra and investigate the ℛ (p , q) -deformed super Virasoro algebra with the particular conformal dimension Δ = 1. Furthermore, we perform the ℛ (p , q) -conformal Virasoro n -algebra, the ℛ (p , q) -conformal super Virasoro n -algebra (n -even) and discuss a toy model for the ℛ (p , q) -conformal Virasoro constraints and ℛ (p , q) -conformal super Virasoro constraints. Besides, we generalized the notion of the ℛ (p , q) -elliptic Hermitian matrix model with an arbitrary conformal dimension Δ. Finally, we deduce relevant particular cases generated by quantum algebras known in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we prove that every matrix over a division ring is representable as a product of at most 10 traceless matrices as well as a product of at most four semi-traceless matrices. By applying this result and the obtained so far other results, we show that elements of some algebras possess some rather interesting and nontrivial decompositions into products of images of non-commutative polynomials. [ABSTRACT FROM AUTHOR]

Let K be a field of characteristic zero and x a free variable. A K - ℰ -derivation of K [ x ] is a K -linear map of the form I , − , ϕ for some K -algebra endomorphism ϕ of K [ x ] , where I denotes the identity map of K [ x ]. In this paper, we study the image of an ideal of K [ x ] under some K -derivations and K - ℰ -derivations of K [ x ]. We show that the LFED conjecture proposed in [W. Zhao, Some open problems on locally finite or locally nilpotent derivations and ℰ -derivations, Commun. Contem. Math. 20(4) (2018) 1750056] holds for all K - ℰ -derivations and all locally finite K -derivations of K [ x ]. We also show that the LNED conjecture proposed in [W. Zhao, Some open problems on locally finite or locally nilpotent derivations and ℰ -derivations, Commun. Contem. Math. 20(4) (2018) 1750056] holds for all locally nilpotent K -derivations of K [ x ] , and also for all locally nilpotent K - ℰ -derivations of K [ x ] and the ideals u K [ x ] such that either u = 0 , or deg u ≤ 1 , or u has at least one repeated root in the algebraic closure of K. As a bi-product, the homogeneous Mathieu subspaces (Mathieu–Zhao spaces) of the univariate polynomial algebra over an arbitrary field have also been classified. [ABSTRACT FROM AUTHOR]

For an ample groupoid and a unit x of , Steinberg constructed the induction and restriction functors between the category of modules over the Steinberg algebra A R () and the category of modules over the isotropy group algebra R x . In this paper, we prove a graded version of these functors and give a description of spectral graded simple modules over the graded Steinberg algebra A R (). As an application, the spectral simple and graded simple modules over the Leavitt path algebra L K (E) are classified. In particular, we show that many of previously known simple and graded simple L K (E) -modules, including the Chen simple modules, are induced from (ungraded or graded) simple modules over isotropy group algebras. [ABSTRACT FROM AUTHOR]

Given a Hopf algebra H , Brzeziński and Militaru have shown that each braided commutative Yetter–Drinfeld H -module algebra A gives rise to an associative A -bialgebroid structure on the smash product algebra A ♯ H. They also exhibited an antipode map making A ♯ H the total algebra of a Lu's Hopf algebroid over A. However, the published proof that the antipode is an antihomomorphism covers only a special case. In this paper, a complete proof of the antihomomorphism property is exhibited. Moreover, a new generalized version of the construction is provided. Its input is a compatible pair A and A op of braided commutative Yetter–Drinfeld H -module algebras, and output is a symmetric Hopf algebroid A ♯ H ≅ H ♯ A op over  A. This construction does not require that the antipode of H is invertible. [ABSTRACT FROM AUTHOR]

In this paper, we shed new light on the Flexible Atom Conjecture. We first give finite representation results for relation algebras 3 3 3 7 , 3 5 3 7 , 7 7 8 3 , 7 8 8 3 , 8 0 8 3 , 8 2 8 3 , 8 3 8 3 , 1 3 1 0 1 3 1 6 , 1 3 1 3 1 3 1 6 , 1 3 1 5 1 3 1 6 and 1 3 1 6 1 3 1 6 . Prior to our paper, only 8 3 8 3 and 1 3 1 6 1 3 1 6 were known to be finitely representable. We accomplish this by generalizing the notion of a relation algebra generated by a Ramsey scheme to the directed (antisymmetric) setting, and then showing that each of these algebras embeds into a finite directed anti-Ramsey scheme. The notion of a directed anti-Ramsey scheme may be of independent interest. We complement our upper bounds with some lower bounds. Namely, we show that any square representation of 3 1 3 7 requires at least 14 points, any square representation of 3 3 3 7 requires at least 11 points, and any square representation of 3 5 3 7 requires at least 12 points. Our technique adapts previous work of Alm et al. [Algebra Univ. (2022)], in that we examine the combinatorial structure induced by the flexible atom. [ABSTRACT FROM AUTHOR]

Random hypergraphs and random simplicial complexes on finite vertices were studied by [M. Farber, L. Mead and T. Nowik, Random simplicial complexes, duality and the critical dimension, J. Topol. Anal.41(1) (2022) 1–32]. The map algebra on random sub-hypergraphs of a fixed simplicial complex, which detects relations between random sub-hypergraphs and random simplicial sub-complexes, was studied by the authors of this paper. In this paper, we study the map algebra on random sub-hypergraphs of a fixed hypergraph. We give some algorithms generating random hypergraphs and random simplicial complexes by considering the actions of the map algebra on the space of probability distributions. We prove some Künneth-type formulae for random hypergraphs and random simplicial complexes on finite vertices. [ABSTRACT FROM AUTHOR]

In an unpublished work of Fasel–Rao–Swan the notion of the relative Witt group W E (R , I) is defined. In this paper, we will give the details of this construction. Then we study the injectivity of the relative Vaserstein symbol V R , I : U m 3 (R , I) / E 3 (R , I) → W E (R , I). We establish injectivity of this symbol if R is a non-singular affine algebra of dimension 3 over a perfect C 1 -field and I is a local complete intersection ideal of R. It is natural to ask whether the Vaserstein symbol is injective for a singular 3-dimensional affine algebra. At the end of the paper, we will give an example of a singular 3-dimensional algebra over a perfect C 1 -field for which the Vaserstein symbol is injective. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of essential extensions of fuzzy modules. We use these concepts to introduce the notion of injective hulls of fuzzy modules. It is known that every R-module has an injective hull, where R is a ring. We show that these corresponding results do not hold for fuzzy R-modules, i.e. there exists a fuzzy R-module that does not have an injective hull. Sufficient conditions are given for a fuzzy R-module to have an injective hull. [ABSTRACT FROM AUTHOR]

This paper explores several topics related to Woodin's HOD conjecture. We improve the large cardinal hypothesis of Woodin's HOD dichotomy theorem from an extendible cardinal to a strongly compact cardinal. We show that assuming there is a strongly compact cardinal and the HOD hypothesis holds, there is no elementary embedding from HOD to HOD, settling a question of Woodin. We show that the HOD hypothesis is equivalent to a uniqueness property of elementary embeddings of levels of the cumulative hierarchy. We prove that the HOD hypothesis holds if and only if every regular cardinal above the first strongly compact cardinal carries an ordinal definable ω -Jónsson algebra. We show that if the HOD hypothesis holds and HOD satisfies the Ultrapower Axiom, then every supercompact cardinal is supercompact in HOD. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce the notion of Rota–Baxter mock-Lie bialgebras (A , ∘ , δ , K , Q) and their admissibility conditions in terms of dual representations (ρ ∗ , β ∗ , V ∗). Next, we show that Rota–Baxter mock-Lie bialgebras are characterized by matched pairs and Manin triples of Rota–Baxter mock-Lie algebras. Furthermore, the coboundary case leads to the introduction of the admissible mock-Lie Yang–Baxter equation in Rota–Baxter mock-Lie algebras, whose skew-symmetric solutions are used to construct Rota–Baxter mock-Lie bialgebras. [ABSTRACT FROM AUTHOR]

Action of a pomonoid on partially ordered sets (S -posets) has beautiful aspects in practical subjects such as automata theory, projection algebra and theoretical computer science which makes it always capture the interest of mathematicians. On the other hand, the study of different kinds of weakly injectivity (which category theory inherited from homological and commutative algebra) is an interesting subject for mathematicians. One of the important kinds of weakly injectivity is quasi-injectivity. In this paper, we study quasi-injectivity in the category of S -posets with respect to special kind of order embeddings, namely, down-closed embeddings (dc-quasi-injectivity). [ABSTRACT FROM AUTHOR]

Operated algebras, that is, algebras equipped with linear operators, have important applications in mathematics and physics. Two primary instances of operated algebras are the Rota–Baxter algebra and TD-algebra. In this paper, we introduce a λ -TD algebra that includes both the Rota–Baxter algebra and the TD-algebra. The explicit construction of free commutative λ -TD algebra on a commutative algebra is obtained by a generalized shuffle product, called the λ -TD shuffle product. We then show that the free commutative λ -TD algebra possesses a left counital bialgebra structure by means of a suitable 1-cocycle condition. Furthermore, the classical result that every connected filtered bialgebra is a Hopf algebra, is extended to the context of left counital bialgebras. Given this result, we finally prove that the left counital bialgebra on the free commutative λ -TD algebra is connected and filtered, and thus is a left counital Hopf algebra. [ABSTRACT FROM AUTHOR]

In this paper, we continue the study of Hurwitz series over a commutative unital ring that was begun in [W. Keigher, On the ring of Hurwitz series, Commun. Algebra 25 (1997) 1845–1859; W. Keigher and F. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000) 291–304]. In particular, we introduce the notion of multivariate Hurwitz series. The underlying idea is that multivariate Hurwitz series are to Hurwitz series as studied in [W. Keigher, On the ring of Hurwitz series, Commun. Algebra 25 (1997) 1845–1859; W. Keigher and F. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000) 291–304] as formal power series in several indeterminates are to formal power series in only one indeterminate. The elementary aspects of the theory follow along the lines of [W. Keigher, On the ring of Hurwitz series, Commun. Algebra 25 (1997) 1845–1859; W. Keigher and F. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000) 291–304]. The treatment of substitution and divided powers introduces special problems not encountered in [W. Keigher and F. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000) 291–304] and requires special attention to subtle details. However, we are able to establish analogous results. With substitution and divided powers in place, we construct and study the analog to the so called inner transformations of [S. Bochner and W. T. Martin, Several Complex Variables (Princeton University Press, 1948)]. Finally, we are able to establish analogs to many of the fundamental results of single and multivariate calculus. [ABSTRACT FROM AUTHOR]

This paper is devoted to defining and studying Whittaker modules and high order Whittaker modules for the N = 1 super-BMS3 algebra. We also classify the simple Whittaker modules and obtain the necessary and sufficient conditions for the irreducibility of these modules. [ABSTRACT FROM AUTHOR]

In this paper, we construct a holomorphic Siegel modular form of weight 2 and level 2, and compute its Fourier coefficients explicitly. Moreover, we prove that this modular form equals the generating function of the representative number ρ (n , m , r) associated with the maximal order in the quaternion algebra (− 1 , − 1) ℚ . As a corollary, we can give a new proof of the famous formula for the sums of three squares. As applications, we give an explicit formula for the numbers of solutions of two systems of Diophantine equations related with Sun's "1-3-5 conjecture". Furthermore, we show that "a perfect square" in the integral condition version of Sun's conjecture can be replaced by "a power of 4". [ABSTRACT FROM AUTHOR]