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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]

The paper is devoted to the so-called complete Leibniz algebras. It is known that a Lie algebra with a complete ideal is split. We will prove that this result is valid for Leibniz algebras whose complete ideal is a solvable algebra such that the codimension of nilradical is equal to the number of generators of the nilradical. [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 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]

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]

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 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]

By measured graphs, we mean graphs endowed with a measure on the set of vertices. In this context, we explore the relations between the appropriate Cheeger constant and Poincaré inequalities. We prove that the so-called Cheeger inequality holds in two cases: when the measure comes from a random walk, or when the measure has a bounded measure ratio. Moreover, we also prove that our measured (asymptotic) expanders are generalised expanders introduced by Tessera. Finally, we present some examples to demonstrate relations and differences between classical expander graphs and the measured ones. This paper is motivated primarily by our previous work on the rigidity problem for Roe 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]

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 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]

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]

The paper introduces the left and right versions of the large class of Drazin inverses in terms of the left and right annihilators in a ring, which are called left-Drazin and right-Drazin inverses. We characterize some basic properties of these one-sided Drazin inverses, and discuss Jacobson's lemma for them. In addition, the relation between the Drazin inverses and these two one-sided inverses is given by means of the spectrum and the operator decomposition. As an application, the left-Drazin and right-Drazin invertibilities in the Calkin algebra are investigated. [ABSTRACT FROM AUTHOR]

A term which is a formal expression defined by variables and operation symbols can be described by tree diagram. The class of terms under which the longest distance from the root to each vertex is equal is called completely expanded. In this paper, we consider the partial many-sorted operation defined on the family of all completely expanded terms of type τ and construct the partial algebra satisfying the clone axioms. The partial operations on arbitrary algebra generated by completely expanded terms and their related structures are obtained. [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, we study nullity- 2 toroidal extended affine Lie algebras in the context of vertex algebras and their ϕ -coordinated modules. Among the main results, we introduce a variant of toroidal extended affine Lie algebras, associate vertex algebras to the variant Lie algebras, and establish a canonical connection between modules for toroidal extended affine Lie algebras and ϕ -coordinated modules for these vertex algebras. Furthermore, by employing some results of Billig, we obtain an explicit realization of a class of irreducible modules for the variant Lie algebras. [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]

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]

Two mathematical fields that concentrate on creating and analyzing structures are algebra and graph theory. There are numerous studies linking algebraic structures like groups, rings, fields and vector spaces with graph theory. Several algebraic graphs have been defined based on the properties of the order of the group and its elements. In this paper, we systematically review the literature on such graphs to understand the research dynamics in the field. [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]

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]

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]

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]

In this paper, we classify the derivations of degree-one generalized Weyl algebras over a univariate Laurent polynomial ring. In particular, our results cover the Weyl–Hayashi algebra, a quantization of the first Weyl algebra arising as a primitive factor algebra of U q + ( 5) , and a family of algebras which localize to the group algebra of the infinite group with generators x and y , subject to the relation x y = y − 1 x. [ABSTRACT FROM AUTHOR]

In this paper, we develop a braid theoretic approach for computing the Kauffman bracket skein module of the lens spaces L (p , q) , KBSM(L (p , q)), for q ≠ 0. For doing this, we introduce a new concept, that of an unoriented braid. Unoriented braids are obtained from standard braids by ignoring the natural top-to-bottom orientation of the strands. We first define the generalized Temperley–Lieb algebra of type B, TL 1 , n , which is related to the knot theory of the solid torus ST, and we obtain the universal Kauffman bracket-type invariant, V , for knots and links in ST, via a unique Markov trace constructed on TL 1 , n . The universal invariant V is equivalent to the KBSM(ST). For passing now to the KBSM(L (p , q)), we impose on V relations coming from the band moves (or slide moves), that is, moves that reflect isotopy in L (p , q) but not in ST, and which reflect the surgery description of L (p , q) , obtaining thus, an infinite system of equations. By construction, solving this infinite system of equations is equivalent to computing KBSM(L (p , q)). We first present the solution for the case q = 1 , which corresponds to obtaining a new basis, ℬ p , for KBSM(L (p , 1)) with (⌊ p / 2 ⌋ + 1) elements. We note that the basis ℬ p is different from the one obtained by Hoste and Przytycki. For dealing with the complexity of the infinite system for the case q > 1 , we first show how the new basis ℬ p of KBSM(L (p , 1)) can be obtained using a diagrammatic approach based on unoriented braids, and we finally extend our result to the case q > 1. The advantage of the braid theoretic approach that we propose for computing skein modules of c.c.o. 3-manifolds, is that the use of braids provides more control on the isotopies of knots and links in the manifolds, and much of the diagrammatic complexity is absorbed into the proofs of the algebraic statements. [ABSTRACT FROM AUTHOR]

In this paper, we prove that if R is an affine algebra of dimension d ≥ 4 over ¯ p and 1 / (d − 1) ! ∈ R , then any unimodular row over R of length d can be mapped to a factorial row by elementary transformations. [ABSTRACT FROM AUTHOR]

Let R be a ring of dimension d ≥ 1 containing ℚ and A = R [ X , Y , Z , W ] (X Y − Z W) . This paper examines the question of efficient generation of ideals in A. [ABSTRACT FROM AUTHOR]

Let and ′ be two ∗-Jordan algebras with identities I and I ′ , respectively, and e a nontrivial ∗-idempotent in . In this paper, we study the characterization of multiplicative ∗-Jordan-type maps. In particular, we provide a characterization in the case of unital prime associative algebra endowed with an involution. [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]

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]

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 evaluate some products of distributions in Colombeau algebra of generalized functions. In the classical theory of Schwartz distributions, multiplication of distributions is not defined for two arbitrary singular distributions. The properties of the Colombeau algebra allow us to calculate products of singular distributions which are not defined in the classical theory. The notion of association in Colombeau algebra of generalized functions allows us the results obtained in this way to be considered as products in the classical theory of distributions. The definition of the Colombeau product of distributions can be considered as generalization of their classical product in Schwartz theory. [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]

An algebra A will be called fluid if for any linearly independent set X consisting of units, the set X − 1 of inverses of the elements of X is also linearly independent. We consider various examples of fluid algebras and study fluid algebras in some specific settings including direct sums, field extensions, quotient rings of the ring of polynomials over a field, and matrix algebras. Pertinent parameters for the study of fluidity are also introduced and studied. [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]

In this paper, we introduce an algebra structure denoted by InvDer algebra in which we twist an algebra with an invertible derivation whose inverse is also a derivation. We define InvDer Lie algebras, InvDer associative algebras, InvDer Zinbiel algebras and InvDer dendriforme algebras. We also study the relations between these structures by using Rota–Baxter operators and endomorphism operators. [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, the discovery of the Jones polynomial will be discussed, emphasizing the way in which it illustrated the remarkable unity between distinct parts of mathematics, each with its own language, but initially without a dictionary. [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 [Finite solvable groups with exactly two nonlinear non-faithful irreducible characters, J. Algebra Appl. 18(5) (2019) 1950091], the first author studied directly indecomposable, solvable groups with exactly two nonlinear non-faithful irreducible characters. In this paper, we study the decomposable case and give a classification of decomposable groups possessing exactly two nonlinear non-faithful irreducible characters. In particular, nilpotent groups with this property are also classified. [ABSTRACT FROM AUTHOR]

This is a continuation of a previous study initiated by the third author on nonlocal vertex bialgebras and smash product nonlocal vertex algebras. In this paper, we study a notion of right H -comodule nonlocal vertex algebra for a nonlocal vertex bialgebra H and give a construction of deformations of vertex algebras with a right H -comodule nonlocal vertex algebra structure and a compatible H -module nonlocal vertex algebra structure. We also give a construction of ϕ -coordinated quasi modules for smash product nonlocal vertex algebras. As an example, we give a family of quantum vertex algebras by deforming the vertex algebras associated to non-degenerate even lattices. [ABSTRACT FROM AUTHOR]

Two presentations of quantum affine superalgebras were introduced by Yamane in [On defining relations of affine Lie superalgebras and affine quantized universal enveloping superalgebras, Publ. Res. Inst. Math. Sci. 35 (1999) 321–390], which were called Drinfeld–Jimbo realization and Drinfeld realization. Drinfeld realization contains infinite sequences of generators and relations. In this paper, we consider the Drinfeld realization of quantum affine superalgebra q ( ̂ m | n) associated to type m | n and define a simple algebra 0 ( ̂ m | n) generated by only a finite part of these sequences of quantum affine superalgebra q ( ̂ m | n). We show that the algebra 0 ( ̂ m | n) is isomorphic to the quantum affine superalgebra q ( ̂ m | n). Using the above isomorphism, we prove there exists an isomorphism between the two realizations. [ABSTRACT FROM AUTHOR]

Cyber-Physical Systems (CPS) are the next generation of intelligent systems that integrate information control devices with physical resources. With increasingly close connections between CPS components and frequent interactions, potential defects grow exponentially, rendering the operating environment of CPS unreliable. Therefore, research on methods and theories to ensure the correctness, safety and reliability of CPS is not only an important research topic but also an inevitable challenge. In this paper, we propose a CPS modeling and verification method based on Architecture Analysis & Design Language (AADL) and process algebra to address this challenge. Due to the continuous, time-constrained, stochastic, uncertain and concurrent characteristics of CPS, this paper considers both flexibility and rigor in the modeling process. We first extend the ability of AADL to describe the multiple characteristics of CPS and propose Hybrid Probability-AADL (HP-AADL). Second, this paper introduces conditional execution, conditional interruption and probability operators into Temporal Calculus of Communication Systems (TCCS) and designs a new formal modeling language Hybrid Probability-Temporal Calculus of Communication Systems (HP-TCCS). However, HP-AADL is a semi-formal modeling language that cannot be directly used for formal verification, it cannot strictly guarantee the quality of the established CPS models, including its functional correctness and safety. Therefore, this paper proposes transformation rules from HP-AADL to HP-TCCS, which enables model checking of CPS models described in HP-AADL within HP-TCCS. Additionally, this paper designs a new formal specification language HPAT-Spatial Temporal Logic (HPAT-STL) based on Probabilistic Computation Tree Logic (PCTL) and Spatial Logic, which characterizes the temporal, probabilistic and spatial properties of CPS model. To achieve formal verification of HP-TCCS model and HPAT-STL formulas, this paper proposes a model checking algorithm HPAT-Model Checking Algorithm (HPAT-MCA). Finally, we discuss a typical CPS example to demonstrate the effectiveness of our proposed method in ensuring correct, safe and reliable CPS. [ABSTRACT FROM AUTHOR]

In this paper, we extend the notion of 2-nil ideal introduced by Yetkin Celikel in [E. Yetkin Celikel, 2-nil ideals of commutative rings, Bulletin of the Belgian Mathematical Society Simon Stevin 28 (3) (2021).] to 2-nil submodule which is a subclass of 2-absorbing primary submodules. Let M be an R -module and N be a proper submodule of M. We say that N is a 2-nil submodule of M if whenever a , b ∈ R , m ∈ M and a b m ∈ N , then a b ∈ Nil (M) or a m ∈ N or b m ∈ N. We study the properties of this concept and establish several characterizations. We also investigate the 2-nil ideals of amalgamation. The obtained results yield new original families of examples of 2-nil ideals. [ABSTRACT FROM AUTHOR]

Let α = (A g , α g) g ∈ be a group-type partial action of a connected groupoid on a ring A = ⊕ z ∈ 0 A z and B : = A ⋆ α be the corresponding partial skew groupoid ring. In the first part of this paper, we investigate the relation of several ring theoretic properties between A and B. For the second part, using that every Leavitt path algebra is isomorphic to a partial skew groupoid ring obtained from a partial groupoid action λ , we characterize when λ is group-type. In such a case, we obtain ring theoretic properties of Leavitt path algebras from the results on general partial skew groupoid rings. Several examples that illustrate the results on Leavitt path algebras are presented. [ABSTRACT FROM AUTHOR]