Idempotency Research Articles

Source of semiprimeness of $\ast$-prime rings

This study constructs a structure $S_{R}^{\ast}$ that had never been studied before and obtained new results by defining a subset $S_{R}^{\ast}$ of $R$ as $S_{R}^{\ast}=\left\{ \left. a\in R\right\vert aRa=aRa^{\ast}=(0)\right\} $ where $\ast$ is an involution and it is called as the source of $\ast$-semiprimeness of $R$. Moreover, it investigates some properties of the subset $S_{R}^{\ast}$ in any ring $R$. Additionally, the relation between the prime radical, which provides the opportunity to work on prime rings, has been studied in many ways, and the set $\mathcal{S}_{R}^{\sigma}$, the motivation of this study, is provided. Furthermore, it is proved that $\mathcal{S}_{R}^{\sigma}=\{0\}$ in the case where the ring $R$ is a reduced ($\sigma$-semiprime) ring and $f(\mathcal{S}_{R}^{\sigma})=\mathcal{S}_{f(R)}^{\sigma}$ under certain conditions for a ring homomorphism $f$. Besides, it is presented that for the idempotent element $e$, the inclusion $e\mathcal{S}_{R}^{\sigma}e\subseteq\mathcal{S}_{eRe}^{\sigma}$ is provided, and for the right ideal (ideal) $I$ of the ring $R$, $\mathcal{S}_{R}^{\sigma}(I)$ is a left semigroup ideal (semigroup ideal) of the multiplicative semigroup $R$. In addition, it is analyzed that the set $\mathcal{S}_{R}^{\sigma}$ is contained by the intersection of all semiprime ideals of the ring $R$.

A discussion of bisexual populations with Wolbachia infection as an evolution algebra

In this paper, Wolbachia infection in a bisexual and diploid population with a fixed cytoplasmic incompatibility rate w and maternal transmission rate d is studied as an evolution algebra. As the cytoplasmic incompatibility (CI) of the population causes deaths in the offspring, the evolution algebra of this model is not baric, and is a dibaric algebra if and only if the cytoplasmic incompatibility rate w is 1 and d = 1 . The idempotent elements are given in terms of d and w. Moreover, this algebra has no absolute nilpotent elements when CI expression w ≠ 1 .

Degree-based topological indices of the idempotent graph of the ring [formula omitted]

Let R be a finite commutative ring with a non-zero identity, and Id(R) be the set of idempotent elements of R. The idempotent graph of R, denoted by GId(R), is a simple undirected graph with all elements of R as vertices, and two distinct vertices u, v are adjacent if and only if u+v∈Id(R). In this paper, we consider the idempotent graph of the ring Zn and investigate some degree-based topological indices, such as the general sum-connectivity index, the general Randić index, the general Zagreb index, and the Sombor index of that graph by considering the M-polynomial of the graph.

Lower spaces of multiplicative lattices

AbstractWe consider some distinguished classes of elements of a multiplicative lattice endowed with coarse lower topologies, and call them lower spaces. The primary objective of this paper is to study the topological properties of these lower spaces, including lower separation axioms and compactness. We characterize lower spaces that exhibit sobriety. Introducing the concept of strongly disconnected spaces, we establish a correlation between strongly disconnected lower spaces and the presence of nontrivial idempotent elements in the corresponding multiplicative lattices. Additionally, we provide a sufficient condition for a lower space to be connected. We prove that the lower space of proper elements is a spectral space; we further explore continuous maps between lower spaces.

Characterizations for convolution lattices based on non-distributive lattices

Recently, De Miguel, Bustince and De Baets have conducted a systematic study on convolution lattices based on distributive lattices. There have been few reports on applying non-distributive lattices to a domain of functions. As a complement to their work, in this paper, we carry out an in-depth investigation of convolution operations of the functions between a non-distributive lattice (domain) and a frame (co-domain). We first present an equivalence characterization between non-distributive lattices and idempotent functions and further show that a subset of the set of idempotent functions is closed under convolution operations. We demonstrate that this subset also is a bisemilattice and satisfies the Birkhoff equation under join- and meet-convolution operations. Finally, we analyze and study the lattice structure related to the obtained algebraic structure.

Weaker forms of increasingness of binary operations and their role in the characterization of meet and join operations

Due to the lack of transitivity of the pseudo-order relation of a proper trellis, its meet and join operations are not increasing. In this paper, we identify weaker forms of increasingness of binary operations that are satisfied by the meet and/or join operation of a trellis or lattice. Some of these forms coincide with the classical notion of increasingness when the trellis is a lattice. Furthermore, we demonstrate the role of these weaker forms in the characterization of the meet and join operations of a trellis, lattice or chain as specific idempotent operations.

On commutators of idempotents

Let T be an operator on a Banach space X that is similar to − T via an involution U. Then,U decomposes the Banach space X as X = X 1 ⊕ X 2 with respect to which decomposition we have U = ( I 1 0 0 − I 2 ) , where I i is the identity operator on the closed subspace X i (i = 1, 2). Furthermore, T has necessarily the form T = ( 0 ∗ ∗ 0 ) with respect to the same decomposition. In this note, we consider the question when T is a commutator of the idempotent P = ( I 1 0 0 0 ) and some idempotent Q on X. We also determine which scalar multiples of unilateral shifts on l p spaces ( 1 ≤ p < ∞ ) are commutators of idempotent operators.

THE CLEANNESS OF THE SUBRINGS OF M_2 (Z_P)

Let be a ring. Ring is said to be a clean ring if every element of R can be expressed as the sum of a unit and an idempotent element. Furthermore, there are r-clean rings. An r-clean ring is a generalization of a clean ring. In an r-clean ring, all of its elements can be represented as the sum of a regular element and an idempotent element. Moreover, strongly r-clean rings were introduced. A strongly r-clean ring is a ring where every element of the ring can be expressed as the sum of a regular and an idempotent element, and the multiplication of that regular and idempotent is commutative. On the other hand, there is a ring of the set of matrices over ring denotes by . In this paper, we will discuss the cleanness properties, especially strongly r-clean of the subring of . The aims of this paper are to find the characteristics of strongly r-clean of the subring of . Here, we assumed that is a ring of matrix over .

Relevant sampling in a reproducing kernel subspace of Orlicz space

In this paper, we aim to provide a general paradigm for dealing with the sampling and random sampling problem in a reproducing kernel subspace of Orlicz space [Formula: see text]. We consider the function space [Formula: see text] as the image of an idempotent integral operator on [Formula: see text], where the integral kernel satisfies certain off-diagonal decay and regularity conditions. The model example of such reproducing kernel subspace of [Formula: see text] includes the finitely generated shift-invariant space and signal space with a finite rate of innovation. We show that a signal in [Formula: see text] can be stably reconstructed from its samples at distinct points separated by a sufficiently small gap. Next, we deduce that the random sampling inequality holds with a high probability for the class of functions in [Formula: see text] concentrated on a cube [Formula: see text], when the samples collected at i.i.d. random points are drawn on [Formula: see text] of order [Formula: see text].

Matsuo algebras in characteristic 2

We extend the theory of Matsuo algebras, which are certain non-associative algebras related to 3-transposition groups, to characteristic 2. The decompositions of our algebras are now induced by nilpotent elements associated to lines in the corresponding Fischer space, rather than idempotent elements associated to points. For many 3-transposition groups, this still gives rise to a Z/2Z-graded fusion law, and we provide a complete classification of when this occurs. In one particular small case, arising from the 3-transposition group Sym(4), the fusion law is even stronger, and the resulting Miyamoto group is an algebraic group Ga2⋊Gm.

OPTICAL ART OF A PLANAR IDEMPOTENT DIVISOR GRAPH OF COMMUTATIVE RING

The idempotent divisor graph of a commutative ring R is a graph with a vertex set in R* = R-{0}, where any distinct vertices x and y are adjacent if and only if x.y = e, for some non-unit idempotent element e^2 = e ∈ R, and is denoted by Л(R). Our goal in this work is to transform the planar idempotent divisor graph after coloring its regions into optical art by depending on the reflection of vertices, edges, and planes on the x or y-axes. That is, we achieve Op art solely through pure mathematics in this paper.

Primary Submaximal Ideals in Commutative Weak Idempotent Rings

Let R be a commutative weak idempotent ring (cWIR, for short) with unity, N and be the set of all nilpotent and idempotent elements of R respectively. In this paper, we study the structure of primary submaximal ideals in R and prove that, if P is a primary submaximal ideal of R and for some , then is a maximal ideal of R and , where .

Idempotent and nilpotent elements in octonion rings over Z

In this paper, we show that the set O/Zp, where p is a prime number, does not form a skew field and discuss idempotent and nilpotent elements in the (finite) ring O/Zp. We provide examples and establish conditions for idempotency and nilpotency. Mathematics Subject Classification (2010): 15A33, 15A30, 20H25, 15A03. Received 27 July 2021; Accepted 14 December 2021

Tropical Modeling of Battery Swapping and Charging Station

We propose and investigate a queueing model of a battery swapping and charging station (BSCS) for electric vehicles (EVs). A new approach to the analysis of the queueing model is developed, which combines the representation of the model as a stochastic dynamic system with the use of the methods and results of tropical algebra, which deals with the theory and applications of algebraic systems with idempotent operations. We describe the dynamics of the queueing model by a system of recurrence equations that involve random variables (RVs) to represent the interarrival time of incoming EVs. A performance measure for the model is defined as the mean operation cycle time of the station. Furthermore, the system of equations is represented in terms of the tropical algebra in vector form as an implicit linear state dynamic equation. The performance measure takes on the meaning of the mean growth rate of the state vector (the Lyapunov exponent) of the dynamic system. By applying a solution technique of vector equations in tropical algebra, the implicit equation is transformed into an explicit one with a state transition matrix with random entries. The evaluation of the Lyapunov exponent reduces to finding the limit of the expected value of norms of tropical matrix products. This limit is then obtained using results from the tropical spectral theory of deterministic and random matrices. With this approach, we derive a new exact formula for the mean cycle time of the BSCS, which is given in terms of the expected value of the RVs involved. We present the results of the Monte Carlo simulation of the BSCS’s operation, which show a good agreement with the exact solution. The application of the obtained solution to evaluate the performance of one BSCS and to find the optimal distribution of battery packs between stations in a network of BSCSs is discussed. The solution may be of interest in the case when the details of the underlying probability distributions are difficult to determine and, thus, serves to complement and supplement other modeling techniques with the need to fix a distribution.

Note on the limit property of triangular norms1

In this article, we characterize triangular norms that have not the limit property, which are applied for exploring the characterizations of function f : [0, 1] → [0, 1] with f ( x ) = lim n → ∞ x T ( n ) for a triangular norm T when the function f is continuous. In particular, we prove that a continuous t-norm T satisfies that f (x) &gt;0 for all x ∈ (0, 1) if and only if 0 is an accumulation point of its non-trivial idempotent elements, and the function f is continuous on [0,1] if and only if T = T M .

The points and localisations of the topos of $M$-sets

In previous papers, we were able to prove that, much like in classical algebraic geometry, it is possible to recover our monoid scheme X from the topos \mathsf{Qcoh}(X) . This was achieved using topos points and localisations of \mathsf{Qcoh}(X) . With this philosophy in mind, the aim of this paper is to study the topos of M -sets for non-commutative monoids, especially their points and localisations. We will classify the points and localisations of M -sets for finite monoids in terms of the idempotent elements of M and idempotent ideals of M , respectively. Some of the results obtained in this paper can already be found in previous works, in direct or indirect forms. See the last part of the introduction.

Some notes on possibilistic randomisation with t-norm based joint distributions in strategic-form games

This article continues the investigation started in [18] on the role of possibilistic mixed strategies in strategic-form games. In this earlier work we assumed, as standard in possibility theory, that joint possibility distributions were computed by combining possibilistic mixed strategies with the minimum t-norm. In this paper, we investigate the consequences of defining joint possibility distributions by using any continuous t-norm, with players' expected utilities based on the Choquet integral. We characterise under which conditions a pair of possibilistic mixed strategies is an equilibrium, generalising the results first presented in [18], and also show that the set of equilibria in possibilistic mixed strategies depends on the set of idempotent elements of a t-norm and not just on the chosen t-norm.

Gamma semigroup whose gamma acts are quasi–injective

In this paper, we interpose the notion tools, Noetherian gamma acts, and γ – idempotent elements, to describe gamma semigroups in which all gamma acts are quasi – injective. We get the following equivalent (i) a gamma semigroup S is Noetherian and each finite direct sum of quasi – injective SΓ-act is quasi – injective (ii) direct sum of quasi – injective SΓ-act is quasi – injective. Also we prove that an γ – monoid is ΓRQI if and only if all right Γ- ideal generated by γ – idempotent element.

Computing Idempotent Elements In 3-Cyclic and 4-Cyclic Refined Neutrosophic Rings of Integers

An element X in a ring R is called idempotent if it equals its square. In this paper, we study the idempotent elements in the 3-cyclic refined neutrosophic ring of integers and 4-cyclic refined neutrosophic ring of integers, where we compute all idempotents in those two rings by solving many different linear Diophantine systems which are generated directly from their the algebraic structure. On the other hand, we use the same Diophantine systems to compute all 2-potent 3-cyclic, and 4-cyclic refined neutrosophic integer elements.

On idempotent elements of dually residuated lattice ordered semigroups

On idempotent elements of dually residuated lattice ordered semigroups