Idempotency Research Articles

On Nil-clean Neutrosophic Rings

A ring is said to be nil-clean if every element of the ring can be written as a sum of an idempotent element and a nilpotent element of the ring. In this paper, we generalize this argument to neutrosophic structure. We introduce the structure of nil-clean neutrosophic ring and some of its elementary properties are presented. Also, we have found the equivalence between classical nil-clean ring R and the corresponding neutrosophic ring R(I), refined neutrosophic ring R(I1, I2), and n-refined neutrosophic ring Rn(I).

The Source of Primeness of Rings

In this study, we define a new concept, i.e., source of primeness of a ring $R$, as $P_{R} := \bigcap_{a\in R} S_{R}^{a}$ such that $S_{R}^{a}:=\{b\in R \mid aRb=(0)\}$. We then examine some basic properties of $P_{R}$ related to the ring’s idempotent elements, nilpotent elements, zero divisor elements, and identity elements. Finally, we discuss the need for further research.

On cardinalities of Green’s equivalence classes in the semigroup of difunctional binary relations

In this paper, we considered the set DX, consisting of all binary relations α ⊆ X × X satisfying (∀x,y,u,v ∈ X) (x,u),(x,v),(y,u)∈α ⇒(y,v) ∈ α. This set is an inverse semigroup under a binary operation defined by xα = yβ−1 ≠ ∅, where xα denotes the set of images of x under α, and yβ−1 denotes the set of pre-images of y under β. Combinatorial results relating to Green’s relations in semigroup are obtained. In particular, we obtained cardinalities of Green’s equivalence classes in the semigroup for the case where X is finite. Also, we obtained the number of idempotent elements in to be equal to , where n = |X| and B(k) is the Bell number defined as the number of partitions of a set of k elements.

Existence of invariant idempotent measures by contractivity of idempotent Markov operators

We prove that the idempotent Markov operator generated by contractive max plus normalized iterated function system (IFS) is also a contractive map w.r.t. natural metrics on the space of idempotent measures. This gives alternative proofs of the existence of invariant idempotent measures for such IFSs.

Random sampling and reconstruction in reproducing kernel subspace of mixed Lebesgue spaces

In this article, we consider the random sampling in the image space of an idempotent integral operator on mixed Lebesgue space . We assume some decay and regularity conditions on the integral kernel and show that the bounded functions in can be approximated by an element in a finite‐dimensional subspace of on . Consequently, we show that the set of bounded functions concentrated on is totally bounded and prove with an overwhelming probability that the random sample set uniformly distributed over is a stable set of sampling for the set of concentrated functions on . Further, we propose an iterative scheme to reconstruct the concentrated functions from their random measurements.

Representations of weakly triangular categories

A new class of locally unital and locally finite dimensional algebras A over an arbitrary algebraically closed field is discovered. Each of them admits an upper finite weakly triangular decomposition, a generalization of a split triangular decomposition. It is established that the category A-lfdmod of locally finite dimensional left A-modules is an upper finite fully stratified category in the sense of Brundan-Stroppel. Moreover, A is semisimple if and only if its centralizer subalgebras associated to certain idempotent elements are semisimple. Furthermore, certain endofunctors are defined so as to give categorical actions of some Lie algebras on the subcategory of A-lfdmod consisting of all objects which have a finite standard filtration.As an application, we study representations of A associated to either cyclotomic Brauer categories or cyclotomic Kauffman categories in details, including explicit criteria on the semisimplicity of A over an arbitrary field, and on A-lfdmod being upper finite highest weight category in the sense of Brundan-Stroppel, and on Morita equivalence between A and direct sum of infinitely many (degenerate) cyclotomic Hecke algebras. Finally, we obtain categorifications of representations of the classical limits of coideal algebras, which come from all integrable highest weight modules of sl∞ or slˆe.

On the cancellation problem for quantales

In this paper, we consider the cancellation problem for direct products in the category of quantales. It is easy to see that quantales are not always cancelable. We show that indecomposable quantales, simple quantales, quantales without zero-divisors and quantales without nontrivial idempotent elements except the top element are cancelable. We also prove that each quantale can be embedded into a (unital) cancelable quantale.

A note on J -spaces and relatively connected subsets

In this note we present an investigation of J -spaces and other types of spaces related to J -spaces. We characterise these spaces using the notion of relatively connected subsets. For a completely regular J -space X, we give a description of the βX-extension of idempotent elements of C(X) and characterise completely regular J -spaces using idempotent elements of some subrings of C(X). A brief study of relatively connected subsets which leads to the definition of C-normal spaces is given. We show that this new class of C-normal spaces contains the class of normal spaces as a proper subclass. Michael proved that βX \\ X is relatively connected in βX if and only if X is a completely regular J -space. We further establish conditions which characterise completely regular J -spaces via some closed subsets of βX.

Reversible ring property via idempotent elements

Abstract Regarding the question of how idempotent elements affect the reversible property of rings, we study a version of reversibility depending on idempotents. In this perspective, we introduce right (resp., left) 𝑒-reversible rings. We show that this concept is not left-right symmetric. Basic properties of right 𝑒-reversibility in a ring are provided. Among others, it is proved that if 𝑅 is a semiprime ring, then 𝑅 is right 𝑒-reversible if and only if it is right 𝑒-reduced if and only if it is 𝑒-symmetric if and only if it is right 𝑒-semicommutative. Also, for a right 𝑒-reversible ring 𝑅, 𝑅 is a prime ring if and only if it is a domain. It is shown that the class of right 𝑒-reversible rings lies strictly between that of 𝑒-symmetric rings and right 𝑒-semicommutative rings. Some extensions of rings are studied in terms of 𝑒-reversibility.

Combinatorial results of collapse for order-preserving and order-decreasing transformations

The full transformation semigroup TnTn is defined to consist of all functions from Xn={1,…,n}Xn={1,…,n} to itself, under the operation of composition. In \cite{JMH1}, for any αα in TnTn, Howie defined and denoted collapse by c(α)=⋃t∈\im(α){tα−1:|tα−1|≥2}c(α)=⋃t∈\im(α){tα−1:|tα−1|≥2}. Let OnOn be the semigroup of all order-preserving transformations and CnCn be the semigroup of all order-preserving and decreasing transformations on XnXn=under its natural order, respectively. Let E(On)E(On) be the set of all idempotent elements of OnOn, E(Cn)E(Cn) and N(Cn)N(Cn) be the sets of all idempotent and nilpotent elements of CnCn, respectively. Let UU be one of {Cn,N(Cn),E(Cn),On,E(On)}{Cn,N(Cn),E(Cn),On,E(On)}. For α∈Uα∈U, we consider the set \imc(α)={t∈\im(α):|tα−1|≥2}\imc(α)={t∈\im(α):|tα−1|≥2}. For positive integers 2≤k≤r≤n2≤k≤r≤n, we define U(k)={α∈U:t∈\imc(α) and |tα−1|=k},U(k,r)={α∈U(k):∣∣⋃t∈\imc(α)tα−1|=r}.U(k)={α∈U:t∈\imc(α) and |tα−1|=k},U(k,r)={α∈U(k):|⋃t∈\imc(α)tα−1|=r}. The main objective of this paper is to determine |U(k,r)||U(k,r)|, and so |U(k)||U(k)| for some values rr and kk.

Baer and quasi-Baer annihilator conditions for nearrings and rings

A ring with unity is called Baer (quasi-Baer) if the left annihilator of each nonempty set (ideal) is generated by an idempotent element. The origins of the class of Baer rings evolved as an abstraction of the strictly algebraic properties of von Neumann algebras. This concept has been extended to nearrings. However in the classes of nearrings and rings without unity, the Baer concept splits into at least four distinct classes and at least eight classes for the quasi-Baer concept (see below). We investigate certain nearring and ring decompositions induced by Baer or quasi-Baer annihilator conditions. Examples are provided to illustrate and delimit our results.

The Fibonacci sequence in the description of maximal discrete Archimedean t-norms

There are many arguments for counting with more than two “truth values”; this allows to imitate human reasoning of facts which are not binary. For theoretical reasons, it is natural to use the whole real interval as the scale. However, this brings practical problems: it is difficult, and even impossible, to represent exact values. Often only a small scale of values suffices to express what we need. Therefore, finite chains are frequently used as domains of fuzzy logical operations. Their representation and manipulation are easy. In this paper, we focus on triangular norms (t-norms). The choice of a finite domain admits some operations (Gödel, Łukasiewicz), while it excludes others (all strict ones, including the product). A disadvantage of the Gödel (minimum) t-norm is that repetition of arguments does not change their meaning. This is often desirable to emphasize the statement. (“Words, words, words!”) Thus we do not consider the Gödel operations sufficient for representation of all fuzzy logical statements in human reasoning. Together with them, we discard all operations with idempotent elements other than 0, 1, thus we restrict attention to Archimedean t-norms (and t-conorms as their duals, not treated explicitly in the sequel). A disadvantage of many Archimedean t-norms is that, when applied to several arguments, the result is very often zero. Then, it gives no clue in comparing the outcomes, e.g., in giving priorities to alternatives evaluated by fuzzy rules. For instance, when the continuous Łukasiewicz t-norm is applied to 5 entries, it is nonzero only on 1/5!=1/120 of the volume of its 5-dimensional domain; in its discrete versions, nonzero results are even more rare. Thus we are interested in Archimedean t-norms with values “as large as possible”, here in the maximal Archimedean t-norms (=those which are not majorized by other Archimedean t-norms). It was shown in previous works that there is an abundance of discrete t-norms; their number grows fast with the number of elements of the underlying chain (no exponential bound seems to be known). There is also an abundance of Archimedean t-norms. In contrast to that, when we counted the number of maximal Archimedean t-norms, it grows asymptotically exponentially with a mild base. What is more interesting, these numbers follow the Fibonacci sequence. We have found a description of maximal Archimedean t-norms and explained also the processes behind their construction and the role of the Fibonacci sequence. These results link the t-norms to a construction using a kind of additive generators, although this could not be used without modifications. We do not describe all Archimedean t-norms and their numbers, but we have completed at least a significant step towards this goal, too.

ON THE SEMIGROUP OF DIFUNCTIONAL BINARY RELATIONS

In this paper, we have examine some properties of elements of the semigroup , where DX, is the set of all binary relations α ⊆ X × X satisfying , (), and is a binary operation on DX defined by () , with xα denoting set of images of x under α, and yβ−1 denoting set of pre-images of y under β. In particular, we showed that in the semigroup there is no distinction between the concepts of reflexive and symmetric relations. We also presented a characterization of idempotent elements in in term of equivalence relations.

Ideal spaces: An extension of structure spaces of rings

Can there be a structure space-type theory for an arbitrary class of ideals of a ring? The ideal spaces introduced in this paper allows such a study and our theory includes (but not restricted to) prime, maximal, minimal prime, strongly irreducible, irreducible, completely irreducible, proper, minimal, primary, nil, nilpotent, regular, radical, principal, finitely generated ideals. We characterise ideal spaces that are sober. We introduce the notion of a strongly disconnected spaces and show that for a ring with zero Jacobson radical, strongly disconnected ideal spaces containing all maximal ideals of the ring imply existence of nontrivial idempotent elements in the ring. We also give a sufficient condition for a spectrum to be connected.

The Necessary and Sufficient Condition of Clean R-Coalgebra e⇀C↼e

Let R be a commutative ring with multiplicative identity and C be a coassociative and counital R-coalgebra with the α-condition. A clean comodules defined based on the cleanness on rings and modules. A C-comodule M is a clean comodule if the endomorphism ring of C-comodule M is clean. A clean R-coalgebra C is a clean comodule over itself i.e., if the endomorphism ring of C as a C-comodule is clean. For an idempotent e ∈ R, there are relations between the cleanness of eRe and R. It’s motivated us to investigate this condition for coalgebra. For any C, we can construct the R-coalgebra e ⇀C↼e where e is an idempotent element of dual algebra of C. Here, we show that the clean conditions of C implies the clean property of e ⇀C↼ e and vice versa.

Some applications of idempotent elements in MV algebras

"In this paper we provide some properties and applications of MV-algebras. We prove that a Fibonacci stationary sequence in an MV-algebra gives us an idempotent element. Moreover, taking into account of the representation of a finite MV-algebra, by using Boolean elements of this algebra, we prove that a Fibonacci sequence in an MV-algebra is always stationary. This result is interesting comparing with the behavior of such a sequence on the group $\left( \mathbb{Z}% _{n},+\right) $, where the Fibonacci sequences are periodic, with the period given by the Pisano period. We also give some examples of finite MV-algebras and the number of their idempotent elements. As an application in Coding Theory, to a Boolean algebra it is attached a binary block code and it is proved that, under some conditions, the converse is also true."

Involutive Uninorm Logic with Fixed Point enjoys finite strong standard completeness

An algebraic proof is presented for the finite strong standard completeness of the Involutive Uninorm Logic with Fixed Point ({{mathbf {IUL}}^{fp}}). It may provide a first step towards settling the standard completeness problem for the Involutive Uninorm Logic ({mathbf {IUL}}, posed in G. Metcalfe, F. Montagna. (J Symb Log 72:834–864, 2007)) in an algebraic manner. The result is proved via an embedding theorem which is based on the structural description of the class of odd involutive FL_e-chains which have finitely many positive idempotent elements.

PT symmetric fermionic field theories with axions: Renormalization and dynamical mass generation

We consider the renormalization properties of non-Hermitian Yukawa theories involving a pseudoscalar (axion) field at or near four dimensions. The non-Hermiticity is $\mathcal{P}\mathcal{T}$ symmetric where $\mathcal{P}$ is a linear operator (such as parity) and $\mathcal{T}$ is an antilinear idempotent operator (such as time reversal). The coupling constants of the Yukawa and quartic scalar coupling terms reflect this non-Hermiticity. The path integral representing the field theory is used to discuss the Feynman rules associated with the field theory. The fixed point structure associated with the renormalization group has $\mathcal{P}\mathcal{T}$ symmetric and Hermitian fixed points. At two loops in the massless theory, we demonstrate the flow from Hermitian to non-Hermitian fixed points. From the one-loop renormalization of a massive Yukawa theory, a self-consistent Nambu--Jona-Lasinio gap equation is established and its real solutions are discussed.

On P -Clean Rings

If all the elements of a ring W can be written as a sum of pure and idempotent elements", the ring is said to be P-clean. In this paper, we introduce the P-clean ring nation and look into some of its fundamental properties, examples, and relationship to the clean ring

Canonical Forms for Reachable Systems over Von Neumann Regular Rings

If (A,B) is a reachable linear system over a commutative von Neumann regular ring R, a finite collection of idempotent elements is defined, constituting a complete set of invariants for the feedback equivalence. This collection allows us to construct explicitly a canonical form. Relations are given among this set of idempotents and various other families of feedback invariants. For systems of fixed sizes, the set of feedback equivalent classes of reachable systems is put into 1-1 correspondence with an appropriate partition of Spec(R) into open and closed sets. Furthermore, it is proved that a commutative ring R is von Neumann regular if and only if every reachable system over R is a finite direct sum of Brunovsky systems, for a suitable decomposition of R.