Algebraic Characterization Research Articles

Subalgebras and Ideals in BCK/BCI-algebras Based on Uni-hesitant Fuzzy Set Theory

In the present paper, the notions of uni-hesitant fuzzy algebras and uni-hesitant fuzzy (closed) ideals in BCK-algebras and BCI-algebras are introduced, and several related properties are investigated. Characterizations of uni-hesitant fuzzy algebras and uni-hesitant fuzzy (closed) ideals are considered, and a new uni-hesitant fuzzy algebra (resp. uni-hesitant fuzzy (closed) ideal) from old one is established. Relations between uni-hesitant fuzzy algebras and uni-hesitant fuzzy (closed) ideals are discussed, and conditions for a uni-hesitant fuzzy ideal to be hesitant closed are provided.

Algebraic Characterization of the Local Craig Interpolation Property

The sole purpose of this paper is to give an algebraic characterization, in terms of a superamalgamation property, of a local version of Craig interpolation theorem that has been introduced and studied in earlier papers. We continue ongoing research in abstract algebraic logic and use the framework developed by Andréka– Németi and Sain.

On algebraic characterization of SSC of the Jahangir’s graph 𝓙 n,m

Abstract In this paper, some algebraic and combinatorial characterizations of the spanning simplicial complex Δs (𝓙 n,m ) of the Jahangir’s graph 𝓙 n,m are explored. We show that Δs (𝓙 n,m ) is pure, present the formula for f-vectors associated to it and hence deduce a recipe for computing the Hilbert series of the Face ring k[Δs (𝓙 n,m )]. Finally, we show that the face ring of Δs (𝓙 n,m ) is Cohen-Macaulay and give some open scopes of the current work.

Some properties of semifinite tracial subalgebras

We obtained some characterizations of Jensen type inequalities in tracial subalgebras and gave some characterizations of subdiagonal algebras of semifinite von Neumann algebras.

Some characterizations of algebras with involution with polynomial growth of their codimensions

Let A be an associative algebra endowed with an involution of the first kind and let denote the sequence of -codimensions of A. In this paper, we are interested in algebras with involution such that the -codimension sequence is polynomially bounded. We shall prove that A is of this kind if and only if it satisfies the same identities of a finite direct sum of finite dimensional algebras with involution , each of which with Jacobson radical of codimension less than or equal to one in . We shall also relate the condition of having polynomial codimension growth with the sequence of cocharacters and with the sequence of colengths. Along the way, we shall show that the multiplicities of the irreducible characters in the decomposition of the cocharacters are eventually constants. Finally, we shall give a classification of the algebras with involution whose -codimensions are at most of linear growth.

Connections between two-universe rough sets and formal concepts

Rough sets and formal concept analysis are two complementary tools during the process of data analysis. Two-universe rough set model is one of generalization of the classical rough sets. In this paper, the connections between two-universe rough sets and formal concepts are discussed. We investigate the relations between two-universe rough sets and the object (attribute) oriented formal concepts. We also establish connections between revised two-universe rough sets and the object (attribute) oriented formal concepts. Meanwhile, relations between algebraic characterizations of two-universe rough sets and formal concepts are revealed. Some properties of two-universe rough sets are examined.

Parallel-In-Time Simulation of Eddy Current Problems Using Parareal

In this contribution the usage of the Parareal method is proposed for the time-parallel solution of the eddy current problem. The method is adapted to the particular challenges of the problem that are related to the differential algebraic character due to non-conducting regions. It is shown how the necessary modification can be automatically incorporated by using a suitable time stepping method. The paper closes with a first demonstration of a simulation of a realistic four-pole induction machine model using Parareal.

Leavitt path algebras: Graded direct-finiteness and graded Σ-injective simple modules

In this paper, we give a complete characterization of Leavitt path algebras which are graded Σ-V rings, that is, rings over which a direct sum of arbitrary copies of any graded simple module is graded injective. Specifically, we show that a Leavitt path algebra L over an arbitrary graph E is a graded Σ-V ring if and only if it is a subdirect product of matrix rings of arbitrary size but with finitely many non-zero entries over K or K[x,x−1] with appropriate matrix gradings. We also obtain a graphical characterization of such a graded Σ-V ring L. When the graph E is finite, we show that L is a graded Σ-V ring ⟺L is graded directly-finite ⟺L has bounded index of nilpotence ⟺ L is graded semi-simple. Examples show that the equivalence of these properties in the preceding statement no longer holds when the graph E is infinite. Following this, we also characterize Leavitt path algebras L which are non-graded Σ-V rings. Graded rings which are graded directly-finite are explored and it is shown that if a Leavitt path algebra L is a graded Σ-V ring, then L is always graded directly-finite. Examples show the subtle differences between graded and non-graded directly-finite rings. Leavitt path algebras which are graded directly-finite are shown to be directed unions of graded semisimple rings. Using this, we give an alternative proof of a theorem of Vaš [33] on directly-finite Leavitt path algebras.

Artinian and noetherian partial skew groupoid rings

Let α={αg:Rg−1→Rg}g∈mor(G) be a partial action of a groupoid G on a (not necessarily associative) ring R and let S=R⋆αG be the associated partial skew groupoid ring. We show that if α is global and unital, then S is left (right) artinian if and only if R is left (right) artinian and Rg={0}, for all but finitely many g∈mor(G). We use this result to prove that if α is unital and R is alternative, then S is left (right) artinian if and only if R is left (right) artinian and Rg={0}, for all but finitely many g∈mor(G). This result applies to partial skew group rings, in particular. Both of the above results generalize a theorem by J. K. Park for classical skew group rings, i.e. the case when R is unital and associative, and G is a group which acts globally on R. We provide two additional applications of our main results. Firstly, we generalize I. G. Connell's classical result for group rings by giving a characterization of artinian (not necessarily associative) groupoid rings. This result is in turn applied to partial group algebras. Secondly, we give a characterization of artinian Leavitt path algebras. At the end of the article, we relate noetherian and artinian properties of partial skew groupoid rings to those of global skew groupoid rings, as well as establish two Maschke-type results, thereby generalizing results by M. Ferrero and J. Lazzarin for partial skew group rings to the case of partial skew groupoid rings.

Elementary equivalence of rings with finitely generated additive groups

We give algebraic characterizations of elementary equivalence between rings with finitely generated additive groups. They are similar to those previously obtained for finitely generated nilpotent groups. Here, the rings are not supposed associative, commutative or unitary.

Relations for Modular Forms from Vertex Operator Algebras

We will give a short reminder for vertex operator algebra notion and corresponding characters. Then we discuss algebraic methods for explicit computation of the partition and correlation functions. We then illustrate general ways to find number theory identities for related modular forms by specific examples of modular form relations arising from our construction. Finally, we present new results concerning identities for prime forms on genus g Riemann surfaces and genus two n-point functions for vertex operator algebra characters.

Pseudo-orbit tracing and algebraic actions of countable amenable groups

Consider a countable amenable group acting by homeomorphisms on a compact metrizable space. Chung and Li asked if expansiveness and positive entropy of the action imply existence of an off-diagonal asymptotic pair. For algebraic actions of polycyclic-by-finite groups, Chung and Li proved that they do. We provide examples showing that Chung and Li’s result is near-optimal in the sense that the conclusion fails for some non-algebraic action generated by a single homeomorphism, and for some algebraic actions of non-finitely generated abelian groups. On the other hand, we prove that every expansive action of an amenable group with positive entropy that has the pseudo-orbit tracing property must admit off-diagonal asymptotic pairs. Using Chung and Li’s algebraic characterization of expansiveness, we prove the pseudo-orbit tracing property for a class of expansive algebraic actions. This class includes every expansive principal algebraic action of an arbitrary countable group.

A note on split extensions of bialgebras

Abstract We prove a universal characterization of Hopf algebras among cocommutative bialgebras over an algebraically closed field: a cocommutative bialgebra is a Hopf algebra precisely when every split extension over it admits a join decomposition. We also explain why this result cannot be extended to a non-cocommutative setting.

Right 2-Engel elements, central automorphisms and commuting automorphisms of Lie algebras

Abstract In the present paper, right 2-Engel elements, central automorphisms and commuting automorphisms of Lie algebras will be studied. For this purpose, first the structure of the set of all right 2-Engel elements of a Lie algebra will be examined and then, by taking advantage of it, a number of interesting results about central and commuting automorphisms of Lie algebras will be presented. Finally, a characterization of Lie algebras for which the set of central automorphisms is trivial or the set of commuting automorphisms is trivial will be given.

Extremal polynomials in stratified groups

We introduce a family of extremal polynomials associated with the prolongation of a stratified nilpotent Lie algebra. These polynomials are related to a new algebraic characterization of abnormal subriemannian geodesics in stratified nilpotent Lie groups. They satisfy a set of remarkable structure relations that are used to integrate the adjoint equations.

An algebraic characterization of language-based opacity in labeled Petri nets

Opacity is a property of discrete event systems (DES) that is related to the possibility of hiding a secret to external observers (the intruders). The secret can be either a system state, or a sequence of events executed by the system itself. When the latter type of secrets is considered, the opacity property is referred to as language-based opacity (LBO). This paper deals with LBO when the DES is modeled by a labeled Petri net. One necessary and sufficient condition to check LBO by solving Integer Linear Programming problems is given; such a condition exploits the algebraic representation of Petri nets. A sufficient condition is then derived, which is less demanding from the computational point of view. The effectiveness of the proposed approach is shown by means of examples.

On logarithmic differential operators and equations in the plane

Let $k$ be a field of characteristic zero. Let $f\in k[x_{0},y_{0}]$ be an irreducible polynomial. In this article, we study the space of polynomial partial differential equations of order one in the plane, which admit $f$ as a solution. We provide algebraic characterizations of the associated graded $k[x_{0},y_{0}]$-module (by degree) of this space. In particular, we show that it defines the general component of the tangent space of the curve $\{f=0\}$ and connect it to the $V$-filtration of the logarithmic differential operators of the plane along $\{f=0\}$.

Fuzzy Ideals of C-Algebras

In this paper, we introduce the notion of fuzzy ideals in -algebras and investigate some of their properties. Mainly, we give an algebraic characterization for fuzzy ideals generated by fuzzy sets. Furthermore, it is proved that the class of fuzzy ideals of a −algebra forms an algebraic lattice.

Identification of black hole horizons using scalar curvature invariants

We introduce the concept of a geometric horizon, which is a surface distinguished by the vanishing of certain curvature invariants which characterize its special algebraic character. We motivate its use for the detection of the event horizon of a stationary black hole by providing a set of appropriate scalar polynomial curvature invariants that vanish on this surface. We extend this result by proving that a non-expanding horizon, which generalizes a Killing horizon, coincides with the geometric horizon. Finally, we consider the imploding spherically symmetric metrics and show that the geometric horizon identifies a unique quasi-local surface corresponding to the unique spherically symmetric marginally trapped tube, implying that the spherically symmetric dynamical black holes admit a geometric horizon. Based on these results, we propose a suite of conjectures concerning the application of geometric horizons to more general dynamical black hole scenarios.

A geometric characterization of the classical Lie algebras

An extremal element x in a Lie algebra g is an element for which the space [x,[x,g]] is contained in the linear span of x. Long root elements in classical Lie algebras are examples of extremal elements.Lie algebras generated by extremal elements lead to geometries on points and lines which can be characterized as root shadow spaces of buildings. In this paper we show that, in case the rank of this building is at least 3, the Lie algebra is (up to isomorphism) uniquely defined by this geometry. This provides us with a geometric characterization of (most of the) classical Lie algebras.