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"Real Normed Algebras Revisited," the last paper of the late Gadi Moran, attempts to reconstruct the discovery of the complex numbers, the quaternions, and the octonions, as well as proofs of their properties, using only what was known to 19th-century mathematicians. In his research, Gadi had discovered simple and elegant proofs of the above-mentioned classical results using only basic properties of the geometry of Euclidean spaces and tools from high school geometry. His reconstructions underline an interesting connection between Euclidean geometry and these algebras, and avoid the advanced machinery used in previous derivations of these results. The goal of this article is to present Gadi's derivations in a way that is accessible to a wide audience of readers. [ABSTRACT FROM AUTHOR]

This work is concerned with the consistency conditions for the equations in max plus algebra. The three classes of the max plus linear equations presented in the paper Shinzawa [Ultra discrete permanent and the consistency of max plus linear equations, Linear Algebra Appl. 506 (2016) 445–477] can be equivalently converted into the corresponding system of inequalities. The equivalence relation between ultra discrete permanent and maximum cycle mean is suggested. Thus, the necessary and sufficient conditions for solvability of the equations are obtained using the maximum cycle mean in this note. [ABSTRACT FROM AUTHOR]

This article argues that papermaking supported cryptanalysis's invention; that cryptology—meaning both cryptography and cryptanalysis—could not be practiced before the sustained production of paper; and that Medieval Arabic cryptology originated in tandem with algebra. Furthermore, this article posits that the regional Islamicate postal service, or thebarīd, was used to relay Medieval Arabic cryptograms and thereby shaped the substance of cryptology. These conclusions stem from examining ninth century to fourteenth century Arabic cryptology as a technology and relating Arabic cryptology to three other technological devices: papermaking in the Middle East, algebra, and thebarīd. Extant documents suggest that cryptology originated in ninth century Baghdad. This is because no cryptanalytical writings are known to exist before this period, and cryptology requires both cryptography and cryptanalysis. However, evidence of Medieval Arabic cryptology exists almost exclusively in practitioners' treatises, not in cryptograms or the working papers of cryptanalysts. To redress this barrier to historical research, I move from text to context, or from the ideas in the treatises to thinking about the technologies these ideas called for. Placing cryptological technology within the wider technological context that inspired, shaped, and confined its development suggests why cryptology originated in the Islamicate world, how Arabic cryptology may have been implemented in practice, and what enabled cryptology's start. I hope that this technological exercise encourages the study of the context in which Medieval Arabic cryptology developed, until further primary sources surface. Furthermore, I intend to demonstrate Medieval Arabic cryptology's relevance to cryptological history and Middle Eastern studies. [ABSTRACT FROM PUBLISHER]

This note demonstrates multiple representations (numerical and graphical) of spreadsheet-based quadratic sequences together with prospective teachers' pencil-paper transformations of these numerical sequences into a corresponding symbolization as algebraic expressions. With the majority of prospective teachers, the experience of school mathematics is one of disaffection. They are in a teacher education programme that offers them certification to teach up to grade nine in the public schools. Their work illustrates processes, such as recognizing and extending patterns, by specializing and generalizing particular functional relationships. Insights gained from various methods used by them, together with the instructional affordances, i.e. the instructional benefits of spreadsheets, represent a curricular continuity between quadratic, numerical sequences and related algebraic expressions and suggest a possible change to the order in which the two are introduced. [ABSTRACT FROM AUTHOR]

To optimize anthropogenic material streams, the production process, as well as the quality of the products, must be known. With knowledge of these requirements, it is possible to use extra applied algorithms—in this case evolutionary algorithms as part of artificial intelligence—for the optimization of these secondary material streams. The benefit of this application is the fast and precise calculation of the local and global optima of the optimizing problem. This calculation method uses the benefits of the biological reproduction by applications of mutation, selection, and recombination to find one of the best results in a huge amount of possible and potential results. For the use of secondary materials in the paper production it could be proven that in spite of high quotes of secondary materials in different paper classes, there are some paper classes in which the amount of secondary material could be raised without losing any quality. [ABSTRACT FROM AUTHOR]

In this note we discuss how a modular triangular origami gift box can be made with six rectangular sheets of printing paper. We also derive a formula for the volume of the box based on high school geometry and algebra. [ABSTRACT FROM AUTHOR]

This paper discusses the problem of marking constraints in a class of partially controllable timed Petri nets. Specifically, these are Timed Event Graphs (TEGs) which include disturbed input transitions and which are subject to token capacity constraints in some paths. Using the Min-Plus algebra formalism, formal approaches are proposed to design causal control laws to guarantee compliance with marking constraints in TEGs. Sufficient conditions are defined to show the existence and synthesis of these controllers. The developed methodologies are illustrated in a case study of a disassembly system with a limited stock of components. [ABSTRACT FROM AUTHOR]

Let $ n \geq ~2 $ n ≥ 2 be a natural number, and denote by $ \mathcal {M}_{n} $ M n the algebra of all $ n \times n $ n × n matrices over an algebraically closed field $ \mathbb {F} $ F of zero characteristic. Let also $ K_{1} $ K 1 and $ K_{2} $ K 2 be two non-empty proper subsets of $ \mathbb {F} $ F . In this paper, we characterize linear maps φ on $ \mathcal {M}_{n} $ M n having the property that, for every $ T \in \mathcal {M}_{n} $ T ∈ M n , the spectrum of T is a subset of $ K_1 $ K 1 if and only if the spectrum of $ \varphi (T) $ φ (T) is a subset of $ K_2 $ K 2 . We obtain a similar caracterization for the case when $ K_{1} $ K 1 and $ K_{2} $ K 2 have both at most n elements, working with the equality of the spectrum to the fixed subsets instead of the inclusion. [ABSTRACT FROM AUTHOR]

In this paper, we present a modality-free pre-rough algebra. Łukasiewicz Moisil algebra and Wajsberg algebra are equivalent under a transformation. A similar type of equivalence exists in our proposed definition and standard definition of pre-rough algebra. We obtain a few modality-free algebras weaker than pre-rough algebra. Furthermore, it is also established that modality-free versions for other analogous structures weaker than pre-rough algebra do not exist. Both Hilbert-type axiomatization and sequent calculi for all proposed algebras are presented. [ABSTRACT FROM AUTHOR]

Over the past 50 years, Nelson algebras have been extensively studied by distinguished scholars as the algebraic counterpart of Nelson's constructive logic with strong negation. Despite these studies, a comprehensive survey of the topic is currently lacking, and the theory of Nelson algebras remains largely unknown to most logicians. This paper aims to fill this gap by focussing on the essential developments in the field over the past two decades. Additionally, we explore generalisations of Nelson algebras, such as N4-lattices which correspond to the paraconsistent version of Nelson's logic, as well as their applications to other areas of interest to logicians, such as duality and rough set theory. A general representation theorem states that each Nelson algebra is isomorphic to a subalgebra of a rough set-based Nelson algebra induced by a quasiorder. Furthermore, a formula is a theorem of Nelson logic if and only if it is valid in every finite Nelson algebra induced by a quasiorder. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the study of 2-local derivations on the Schrödinger-Virasoro Algebra, which is an infinite-dimensional Lie algebra with three outer derivations. We prove that all 2-local derivations on the Schrödinger-Virasoro Algebra are derivations. [ABSTRACT FROM AUTHOR]

In this paper, we give a complete classification of all free $ U(\mathbb {C}L_0 \oplus \mathbb {C}Y_0\oplus \mathbb {C}M_0) $ U (C L 0 ⊕ C Y 0 ⊕ C M 0) -modules of rank 1 over a Schrödinger-Virasoro type algebra $ \mathfrak {tsv} $ tsv . [ABSTRACT FROM AUTHOR]

The present paper is devoted to studying local derivations on the Lie algebra $ W(2,2) $ W (2 , 2) which has some outer derivations. Using some linear algebra methods in [1] and a key construction for $ W(2,2) $ W (2 , 2) , we prove that every local derivation on $ W(2, 2) $ W (2 , 2) is a derivation. As an application, we determine all local derivations on the deformed $ \mathfrak {bms}_3 $ b m s 3 algebra. [ABSTRACT FROM AUTHOR]

In this work, we study the relational representation of the class of Tarski algebras endowed with a subordination, called subordination Tarski algebras. These structures were introduced in a previous paper as a generalisation of subordination Boolean algebras. We define the subordination Tarski spaces as topological spaces with a fixed basis endowed with a closed relation. We prove that there exist categorical dualities between categories whose objects are subordination Tarski algebras and categories whose objects are subordination Tarski spaces. These results extend the known dualities for modal algebras. Finally, we are going to characterise some algebraic conditions written in the quasi-modal language by means of first-order conditions. [ABSTRACT FROM AUTHOR]

In this note we describe the mathematics that emerges from the construction of an origami box. We first construct a simple origami box from two rectangular sheets and then discuss some of the mathematical questions that arise in the context of algebra, geometry and calculus. [ABSTRACT FROM AUTHOR]

Fostering students' relational understanding of the equals sign is a challenge for math educators that begins in the primary levels and persists into tertiary education. In this paper, I develop an entry point, especially for students who only have an operational understanding of the equals sign, to the core idea of equivalence in linear algebra. My approach is informed by the history of mathematics: In the 17th and 18th centuries, mathematics research underwent an algebraicization, with mathematicians replacing their classical geometric questions with novel algebraic investigations. In this paper, I will offer geometric interpretations of two operations developed at the precipice of this monumental shift: algebraic substitution and Gaussian elimination. I will then utilize Lakoff & Johnson's theory of conceptual metaphor to compare and contrast this historically-grounded geometric re-interpretation of modern linear algebra to the direct algebraic interpretation taken in most modern textbooks. [ABSTRACT FROM AUTHOR]

Two groups of mathematics faculty, one from a four-year college and one from a two-year college, redesigned their respective introductory college mathematics courses following presentation of observational data regarding how faculty had been teaching the courses. This presentation emphasised how infrequently faculty teaching introductory college mathematics employed recommended pedagogical practices. This work was part of a multi-year federal grant project designed to increase the numbers of underrepresented students majoring in STEM disciplines. While the two teams developed very different redesigned course activities, in both instances the primary motivation for initiating the work was the information provided to faculty in a professional development workshop regarding how they had previously been observed teaching the mathematics content and how infrequently they utilised the pedagogical practices recommended by STEM education experts. The paper also highlights faculty resistance to curricular reform and enumerates some ways of addressing this resistance. The paper also discusses why faculty from two-year and four-year institutions resisted working together on course redesign and provides recommendations for future efforts addressing course redesign. [ABSTRACT FROM AUTHOR]

The generalized L'vov–Kaplansky conjecture states that for any finite-dimensional simple algebra A the image of a multilinear polynomial on A is a vector space. In this paper, we prove it for the algebra of octonions $ \mathbb {O} $ O over a field F satisfying certain specified conditions (in particular, we prove it for quadratically closed fields, and for the field $ \mathbb R $ R ). In fact, letting V be the space of pure octonions in $ \mathbb {O} $ O , we prove that the image set must be either $ \{0\} $ { 0 } , F, V or $ \mathbb {O} $ O . We discuss possible evaluations of semihomogeneous polynomials on $ \mathbb {O} $ O and of arbitrary polynomials on the corresponding Malcev algebra. [ABSTRACT FROM AUTHOR]

Köthe's classical problem posed by G. Köthe in 1935 asks to describe the rings R such that every left R-module is a direct sum of cyclic modules (these rings are known as left Köthe rings). Köthe, Cohen and Kaplansky solved this problem for all commutative rings (that are Artinian principal ideal rings). During the years 1962 to 1965, Kawada solved Köthe's problem for basic finite-dimensional algebras. But, so far, Köthe's problem was open in the non-commutative setting. Recently, in the paper [Several characterizations of left Köthe rings, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. (2023) (to appear)]], we classified left Köthe rings into three classes one contained in the other: left Köthe rings, strongly left Köthe rings and very strongly left Köthe rings, and then, we solved Köthe's problem by giving several characterizations of these rings in terms of describing the indecomposable modules. In this paper, we will introduce the Morita duals of these notions as left co-Köthe rings, strongly left co-Köthe rings and very strongly left co-Köthe rings, and then, we give several structural characterizations for each of them. [ABSTRACT FROM AUTHOR]

The notion of the weighted core inverse in a ring with involution was introduced recently [Mosić et al. Comm. Algebra, 2018; 46(6); 2332–2345]. In this paper, we explore new characterizations of the weighted core inverse of sum and the difference between two weighted core invertible elements in a ring with involution under different conditions. Further, we discuss reverse order and mixed-type reverse order laws for the weighted core invertible elements in a ring. [ABSTRACT FROM AUTHOR]

Quasi-Nelson logic (QNL) was recently introduced as a common generalisation of intuitionistic logic and Nelson's constructive logic with strong negation. Viewed as a substructural logic, QNL is the axiomatic extension of the Full Lambek Calculus with Exchange and Weakening by the Nelson axiom, and its algebraic counterpart is a variety of residuated lattices called quasi-Nelson algebras. Nelson's logic, in turn, may be obtained as the axiomatic extension of QNL by the double negation (or involutivity) axiom, and intuitionistic logic as the extension of QNL by the contraction axiom. A recent series of papers by the author and collaborators initiated the study of fragments of QNL, which correspond to subreducts of quasi-Nelson algebras. In the present paper we focus on fragments that contain the connectives forming a residuated pair (the monoid conjunction and the so-called strong Nelson implication), these being the most interesting ones from a substructural logic perspective. We provide quasi-equational (whenever possible, equational) axiomatisations for the corresponding classes of algebras, obtain twist representations for them, study their congruence properties and take a look at a few notable subvarieties. Our results specialise to the involutive case, yielding characterisations of the corresponding fragments of Nelson's logic and their algebraic counterparts. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to describe the mathematics that emanates from the construction of an origami box. We first construct a simple origami box from a rectangular sheet and then discuss some of the mathematical questions that arise in the context of geometry and algebra. The activity can be used as a context for illustrating how algebra and geometry, like other branches of mathematics, are interrelated. [ABSTRACT FROM AUTHOR]

This work is devoted to the complete classification of solvable Leibniz algebras with the naturally graded quasi-filiform non-Lie nilradical. It should be noted that the class of naturally graded quasi-filiform Leibniz algebras is sufficiently large, and solvable Leibniz algebras with the nilradical which is isomorphic to several of these algebras are classified in a series of works. This paper considered all omitted cases and completed the problem of classifying solvable Leibniz algebras with naturally graded quasi-filiform non-Lie nilradical. At the end of the paper we combine all results obtained to date, and provide a complete list of solvable Leibniz algebras, whose nilradical is a naturally graded quasi-filiform Leibniz algebra. [ABSTRACT FROM AUTHOR]

In this paper, we construct a class of non-weight modules over the affine-Virasoro algebra of type A1 by taking the tensor product of irreducible U(h)-free modules of rank 1 with irreducible highest weight modules. The irreducibility and the isomorphism classes of these modules are determined. Moreover, we show that these tensor product modules are different from the known non-weight modules. Finally, we realize some tensor product modules as induced modules from modules over certain subalgebras of the affine-Virasoro algebra of type A1, and give sufficient and necessary conditions for these induced modules to be reducible. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the relationships between a Hom-Jordan algebras and its induced ternary Hom-Jordan algebras. We give some properties of the αk-generalized derivation algebra G D e r (J) of a ternary Hom-Jordan algebras. In particular, we prove that G D e r (J) = Q D e r (J) + Q Γ (J) , the sum of the αk-quasiderivation algebra and the αk-quasicentroid. We also prove that Q D e r (J) can be embedded as derivations in a larger ternary Hom-Jordan algebras. General results on αk-centroids of ternary Hom-Jordan algebras are also developed in the paper. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the structure of the variety of Lie algebras with a finite number of ideals and their graph representations using Hasse diagrams. The large number of necessary conditions on the algebraic structure of this type of algebras leads to the explicit description of those algebras in the variety with trivial Frattini subalgebra. To illustrate our results, we have included and discussed lots of examples throughout the paper. [ABSTRACT FROM AUTHOR]

Competence in fractions is important in achieving advanced mathematics such as algebra and calculus. To foster students' understanding of fractions, intervention studies have found number lines to be an effective representational tool. Yet, it is unclear whether or not number lines are equally effective regardless of the ways in which they are implemented. The present study compared two ways in which number lines were used to teach fourth graders fraction magnitudes. One was an iPad digital game and the other was a paper-and-pencil workbook. The students in the digital-game condition played the game on their own whereas those in the workbook condition received one-on-one tutoring. Taking part in just four lessons, both groups improved their fraction knowledge compared to the control group. The current findings confirm that number lines are an effective representational tool. The results further suggest that number lines – when implemented as a digital game or in a workbook – are on the average equally effective in improving students' fraction knowledge. Implications for classroom instruction are discussed. [ABSTRACT FROM AUTHOR]

Recently, relative Rota–Baxter (Lie/associative) algebras are extensively studied in the literature from cohomological points of view. In this paper, we consider relative Rota–Baxter Leibniz algebras (rRB Leibniz algebras) as the object of our study. We construct an $ L_\infty $ L ∞ -algebra that characterizes rRB Leibniz algebras as its Maurer–Cartan elements. Then we define representations of an rRB Leibniz algebra and introduce cohomology with coefficients in a representation. As applications of cohomology, we study deformations and abelian extensions of rRB Leibniz algebras. [ABSTRACT FROM AUTHOR]

The tensor product non-weight Virasoro modules M (V , λ 0) ⊗ ⨂ i = 1 m Ω (λ i , α i) are studied in this paper, where Ω (λ i , α i) and M (V , λ 0) (with λ 0 , λ i , α i ∈ C ∗ and V being an irreducible module over a certain finite dimensional Lie algebra related to the Virasoro algebra) are irreducible Virasoro modules, respectively, defined in Liu and Zhao [Generalized polynomial modules over the Virasoro algebra. Proc Amer Math Soc. 2016;144:5103–5112] and Lü and Zhao [Irreducible Virasoro modules from irreducible Weyl modules. J Algebra. 2014;414:271–287]. The necessary and sufficient condition for M (V , λ 0) ⊗ ⨂ i = 1 m Ω (λ i , α i) to be irreducible is obtained. Then we determine the necessary and sufficient condition for two such irreducible modules to be isomorphic. At last, we show that the irreducible L -modules M (V , λ 0) ⊗ ⨂ i = 1 m Ω (λ i , α i) are new for m>1. [ABSTRACT FROM AUTHOR]

Max–min algebra is defined as a linearly ordered set with two binary operations. Classical addition and multiplication are replaced by maximum and minimum, respectively. A square matrix is called robust, if all its orbits converge to an eigenvector. In practice, matrix inputs and start vector inputs are rather contained in some intervals than exact values. Considering matrices and vectors with interval coefficients is therefore of great practical importance. In this paper, we study a special type of the robustness called the X -robustness, the case when a starting vector is limited by a lower bound vector x _ and an upper bound vector x ¯ . If we consider some components of the interval starting vector with the existential quantifier and others with the general quantifier, two new types of X -robustness arise, called X E A -robustness and X A E -robustness. Using a similar quantification of interval matrix elements, we obtain AE/EA- X E A -robustness and AE/EA- X A E -robustness. A complete characterization of AE/EA- X A E -robustness and AE/EA- X E A -robustness for a special type of interval matrices, so-called interval circulant matrices, is presented. [ABSTRACT FROM AUTHOR]

Let A be an unital algebra over the complex field C . A linear map ϕ from A into itself is called a Lie centralizer at a given point G ∈ A if ϕ ([ S , T ]) = [ S , ϕ (T) ] = [ ϕ (S) , T ] for all S , T ∈ A with ST = G. The aim of this paper is to give a description of Lie centralizers at an arbitrary but fixed point on triangular algebras. These results are then applied to nest algebras and upper triangular matrix algebras. [ABSTRACT FROM AUTHOR]

Pretabularity is the attribute of logics that are not characterised by finite matrices, but all of whose proper extensions are. Two of the first-known pretabular logics were Dummett's famous super-intuitionistic logic LC and the well-known semi-relevance logic RM (= R-Mingle). In this paper, we investigate Anderson and Belnap's relevance logic R with the extra axiom: (p → q) ∨ (q → p), which we name LR, and which is (much) weaker than RM, and so is not pretabular. This means that over LR, there may be some pretabular extensions other than RM, two of which this paper presents and with which it provides axiomatizations, characteristic algebras, and Routley-Meyer semantics. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the description of local derivations on solvable Lie algebras of maximal rank. Namely, we consider a solvable Lie algebra of the form R = Q ⊕ N , where Q is the maximal torus subalgebra of R , N is the nilradical of R and dim Q = dim N / N 2. We prove that any local derivation of such solvable Lie algebra R is a derivation. Further, we present two examples of solvable Lie algebras which satisfy the condition dim Q < dim N / N 2 and the first algebra admit a local derivation which is not a derivation, while for the second algebra we prove that any local derivation is a derivation. We also apply the main result of the paper to the description of local derivations on so-called standard Borel subalgebras of complex simple Lie algebras. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a technology-enhanced pedagogical sequence for supporting lower secondary school students' sense-making of the concept of volume in a non-procedural and non-formula-driven way. Specifically, we illustrate a novel approach of using dynamic geometric environment (DGE) to introduce the meaning of volume and then deriving the volume of a hemisphere with minimal algebra but simple trigonometry. This method aims to support students' geometrical reasoning through comparing the volume of different three-dimensional shapes and analysing their cross sections two-dimensionally, thus drawing on the concept of area. We highlight the affordances of the designed DGE sketches for fostering a dynamic conception of volume. [ABSTRACT FROM AUTHOR]

AbstractLeibniz algebras are certain generalizations of Lie algebras. Motivated by the concept of subinvariance and ascendancy in group theory, Schenkman studied properties of subinvariant subalgebras of a Lie algebra. Kawamoto studied ascendency in Lie algebras. In this paper we define subinvariance and asendency in Leibniz algebras and study their properties. It is shown that the signature results on subinvariance and ascendency in Lie algebras have analogs for Leibniz algebras. [ABSTRACT FROM AUTHOR]

Making connections within and between different aspects of mathematics is recognised as fundamental to learning mathematics with understanding. However, exactly what these connections are and how they serve the goal of learning mathematics is rarely made explicit in curriculum documents with the result that mathematics tends to be presented as a set of discrete, disconnected topics. Interest in establishing a more coherent approach to the teaching and learning of school mathematics has led to a focus on big ideas. That is, networks of related concepts, skills and ways of thinking that facilitate learning mathematics with understanding. Research on learning progressions has helped identify what these big ideas are and how they serve to build connections within and between different aspects of mathematics. This paper draws on research that provides an evidenced-based learning progression for multiplicative reasoning to illustrate the connective role of multiplicative thinking in the development of algebraic, geometrical, and statistical reasoning. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we introduce a combinatorial path model of representation of the quantum affine algebra of type Dn, inspired by Mukhin and Young’s combinatorial path models of representations of the quantum affine algebras of types An and Bn. In particular, we give a combinatorial formula for q-characters of fundamental modules of type Dn by assigning each path to a monomial or binomial. By counting our paths, a new expression on dimensions of fundamental modules of type Dn is obtained. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we extend the notion of anti-pre-Lie algebras to the ℤ2-graded version, and introduce the notion of anti-pre-Lie superalgebras. They can be characterized as a class of Lie-admissible superalgebras that satisfy its negative left multiplication operators are the representations of the corresponding sub-adjacent Lie superalgebras. And we give the classification of 2-dimensional anti-pre-Lie superalgebras. We introduce the notion of anti-super -operators on Lie superalgebras to explore the relationships between anti-super -operators and anti-pre-Lie superalgebras. We show that nondegenerate super-commutative 2-cocycles on Lie superalgebras can obtain a class of compatible anti-pre-Lie superalgebra structures. In addition, we introduce a subclass of anti-pre-Lie superalgebras, namely admissible Novikov superalgebras, which correspond to Novikov superalgebras through q-superalgebras. Finally, we introduce the notions of anti-pre-Lie Poisson superalgebras and admissible Novikov-Poisson superalgebras, extending the correspondence to the level of Poisson type structures, the correspondence of the Novikov-Poisson superalgebras and the admissible Novikov-Poisson superalgebras are realized through admissible pairs. [ABSTRACT FROM AUTHOR]

In this paper, first we give the controlling algebra of Lie triple systems. In particular, the cohomology of Lie triple systems can be characterized by the controlling algebra. Then using controlling algebras, we introduce the notions of homotopy Nambu algebras and homotopy Lie triple systems. We show that 2-term homotopy Lie triple systems is equivalent to Lie triple 2-systems, and the latter is the categorification of a Lie triple system. Finally we study skeletal and strict Lie triple 2-systems. We show that skeletal Lie triple 2-systems can be classified the third cohomology group, and strict Lie triple 2-systems are equivalent to crossed modules of Lie triple systems. [ABSTRACT FROM AUTHOR]

The notion of pre-Leibniz algebras was recently introduced in the study of Rota-Baxter operators on Leibniz algebras. In this paper, we first construct a graded Lie algebra whose Maurer-Cartan elements are pre-Leibniz algebras. Using this characterization, we define the cohomology of a pre-Leibniz algebra with coefficients in a representation. This cohomology is shown to split the Loday-Pirashvili cohomology of Leibniz algebras. As applications of our cohomology, we study formal and finite order deformations of a pre-Leibniz algebra. Finally, we define homotopy pre-Leibniz algebras and classify some special types of homotopy pre-Leibniz algebras. [ABSTRACT FROM AUTHOR]

This paper is a first step toward the full description of a family of Hopf algebras whose coradical is isomorphic to a semisimple Hopf algebra Kn, n an odd positive integer, obtained by a cocentral abelian cleft extension. We describe the simple Yetter-Drinfeld modules, compute the fusion rules and determine the finite-dimensional Nichols algebras for some of them. In particular, we give the description of the finite-dimensional Nichols algebras over simple modules over K3. This includes a family of 12-dimensional Nichols algebras { B ξ } depending on 3rd roots of unity. Here, B 1 is isomorphic to the well-known Fomin-Kirillov algebra, and B ξ ≃ B ξ 2 as graded algebras but B 1 is not isomorphic to B ξ as algebra for ξ ≠ 1 . As a byproduct we obtain new Hopf algebras of dimension 216. [ABSTRACT FROM AUTHOR]

AbstractPre-Novikov algebras and quasi-Frobenius Novikov algebras naturally appear in the theory of Novikov bialgebras. In this paper, we show that there is a natural pre-Novikov algebra structure associated to a quasi-Frobenius Novikov algebra. Then we introduce the definition of double constructions of quasi-Frobenius Novikov algebras associated to two pre-Novikov algebras and show that it is characterized by a pre-Novikov bialgebra. We also introduce the notion of pre-Novikov Yang-Baxter equation, whose symmetric solutions can produce pre-Novikov bialgebras. Moreover, the operator forms of pre-Novikov Yang-Baxter equation are also investigated. [ABSTRACT FROM AUTHOR]

AbstractIn this paper we study certain combinatorial attributes of Ind-schemes of polynomial automorphisms in positive characteristic. In particular, we prove that over an algebraically closed field K of positive characteristic ≠2 every automorphism of the group of origin-preserving automorphisms of the polynomial algebra K[x1,…,xn] (n≥3), which fixes every diagonal matrix, preserves, up to composition with a linear inner automorphism, every tame automorphism. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a basis for the center of the infinitesimal q-Schur algebra s q (2 , r) 1 in terms of its certain basis. [ABSTRACT FROM AUTHOR]

AbstractFor a simple graph G with edge ideal I(G), we study the N -graded Betti numbers in the linear strand of the minimal free resolution of I(Γ(Zn)), where Γ(Zn) is the zero divisor graph of the ring Zn . We present sharp bounds for the Betti numbers of Γ(Zn) and characterize the graphs attaining these bounds, thereby establishing the correct equality case for one of the results of the earlier published paper (Theorem 4.5, S. Pirzada and S. Ahmad, On the linear strand of edge ideals of some zero divisor graphs, Commun. Algebra 51(2) (2023) 620–632). Also, we present homological invariants of the edge rings of Γ(Zn) for n=p2q and pqr, with primes p

AbstractIn this paper, all super-skewsymmetric super-biderivations of twisted N = 1 Schrödinger-Neveu-Schwarz algebra are determined. It is shown that all the super-biderivations are inner. Furthermore, the linear super-commuting maps of twisted N = 1 Schrödinger-Neveu-Schwarz algebra are standard. [ABSTRACT FROM AUTHOR]

AbstractThe aim of this paper is to examine the structure of genetic algebras associated with permuted Lotka-Volterra operators on a finite-dimensional simplex. Furthermore, three dimensional permuted Lotka-Voltarra algebras are considered and the description of their derivations are given as well. [ABSTRACT FROM AUTHOR]

Let R be a (semi-) prime ring with extended centroid C, let f(X1, ... , Xk) be a multilinear polynomial over C in k noncommutative indeterminates which is not central-valued on R and let g be a generalized derivation of R. In this paper, we completely characterize the form of g and the structure of R such that (g(f(x1, ... , xk))m − γf(x1, ... , xk)n)s = 0 for all x1, ... , xk ∈ R, where γ ∈ C and m, n, s are fixed positive integers. Our results naturally improve and generalize the theorems obtained by Huang and Davvaz in [Generalized derivations of rings and Banach algebras, Comm. Algebra (2013); 43, 1188–1194] and the theorems recently obtained by De Filippis et al. in [Generalized derivations with nilpotent, power-central and invertible values in prime and semiprime rings, Comm. Algebra (2019); 47, 3025–3039]. Moreover, we describe a revised version of the theorem obtained by Huang in [On generalized derivations of prime and semiprime rings, Taiwanese J. Math. (2012); 16, 771–776.] [ABSTRACT FROM AUTHOR]

This paper aims to study the local derivations, 2-local automorphisms and local automorphisms on the super-Virasoro algebras. The primary focus is to establish that every local derivation of the super-Virasoro algebras is indeed a derivation, and to demonstrate that every local or 2-local automorphism of the super-Virasoro algebras is an automorphism. [ABSTRACT FROM AUTHOR]

For a commutative ring R and a Hopf algebra H which is finitely generated projective as an R-module, it is established that there is an (anti)-isomorphism of groups between the Brauer group BQ(R, H) of Hopf Yetter-Drinfel'd H-module algebras and the Brauer group BQ (R , H *) of Hopf Yetter-Drinfel'd H * -module algebras, where H * is the linear dual of H. In this paper, we generalize this result by establishing an anti-isomorphism of groups between BQ(S, H), the Brauer group of dyslectic Hopf Yetter-Drinfel'd (S, H)-module algebras and BQ (S o p , H *) , the Brauer group of dyslectic Hopf Yetter-Drinfel'd (S o p , H *) -module algebras, where S is an H-commutative Hopf Yetter-Drinfel'd H-module algebra and Sop is the opposite algebra of S. Communicated by Alberto Elduque [ABSTRACT FROM AUTHOR]