Idempotency Research Articles

Metric aggregation functions revisited

In 1981, J. Borsík and J. Doboš analyzed what properties a function must satisfy in order to merge a collection of metric spaces into a single one. Later on, E. Castiñeira, A. Pradera and E. Trillas studied a variant of the same problem in which each metric of the collection to be merged is defined on the same non-empty set. In this, paper we continue the work in this last aforesaid direction. On the one hand, we provide a new characterization of such functions and a few methods to construct them. On the other hand, we analyze the existence of absorbent, idempotent and neutral elements for such class of functions and, thus, we design techniques that allow to discard those functions that are not useful for merging metrics. Finally, we discuss when the functions under study preserve metrics.

The idempotent-divisor graphs of a commutative ring

ABSTRACTGiven an idempotent element e of a commutative ring R, define the idempotent-divisor graph Γe(R) of R associated with e to be the (undirected) graph with vertex-set V(Γe(R)) = {a∈R | there exists b∈R with ab = e} such that distinct a,b∈V(Γe(R)) are adjacent if and only if ab = e. In this paper, the interplay between the algebraic properties of R and the graph-theoretic structure of Γe(R) is investigated. For example, it is shown that if Re is a Boolean ring then Γe(R) is connected, and if R has identity then Γe(R) is finite if and only if R(1−e) and U(R) are finite, where U(R) is the group of units of R. Furthermore, if e∈R and Γe(T(R)) is connected then , where T(R) is the total quotient ring of R. Emphasis is given to linking the structure of Γe(R) with that of zero-divisor graphs (i.e., idempotent-divisor graphs associated with 0).

Idempotents in the group algebra of dihedral groups

In this paper we give the explicit expressions for the idempotent elements in the group algebra of dihedral group of order 2n, for every n.

Derived equivalences and stable equivalences of Morita type, II

We consider the question of lifting stable equivalences of Morita type to derived equivalences. One motivation comes from an approach to Broué’s abelian defect group conjecture. Another motivation is a conjecture by Auslander and Reiten on stable equivalences preserving the number of non-projective simple modules. A machinery is presented to construct lifts for a large class of algebras, including Frobenius-finite algebras introduced in this paper. In particular, every stable equivalence of Morita type between Frobenius-finite algebras over an algebraically closed field can be lifted to a derived equivalence. Consequently, the Auslander–Reiten conjecture is true for stable equivalences of Morita type between Frobenius-finite algebras. Examples of Frobenius-finite algebras are abundant, including representation-finite algebras, Auslander algebras, cluster-tilted algebras and certain Frobenius extensions. As a byproduct of our methods, we show that, for a Nakayama-stable idempotent element e in an algebra A over an algebraically closed field, each tilting complex over eAe can be extended to a tilting complex over A that induces an almost ν-stable derived equivalence studied in the first paper of this series. Moreover, the machinery is applicable to verify Broué’s abelian defect group conjecture for several cases mentioned in the literature.

Rank-metric codes as ideals for subspace codes and their weight properties

Let q=pr, p a prime, r a positive integer, and Fq the Galois field with cardinality q and characteristic p. In this paper, we study some weight properties of rank-metric codes and subspace codes. The rank weight is not egalitarian nor homogeneous, and the rank weight distribution of M2(Fq) is completely determined by the general linear group GL(2,q). We consider subspace weight that is defined on subspace codes and examine their egalitarian property. We also present some examples of rank-metric codes endowed with the rank distance and Grassmannian codes endowed with the subspace distance. These codes were generated from left ideals of M2(Fp) using idempotent elements of M2(Fp).

On the Discrete Bisymmetry

It is known that bisymmetry generalizes the simultaneous commutativity and associativity in the framework of the unit interval. In this work, we will completely characterize two classes of bisymmetric aggregation operators: one with a neutral element and the other with the vertical and horizontal sections of the idempotent elements being smooth on a finite chain, but not necessarily smooth and commutative. Thus, the previous results, based on the smoothness that is known as a very restrictive condition, are improved. For example, there is only one smooth Archimedean t-norm on a finite chain. In this paper, the discrete bisymmetric aggregation operators are explored without the limit of the smoothness. As a by-product, it is deduced that for smooth aggregation operators on a finite chain, the bisymmetry is equivalent to the commutativity and associativity, which improves the conclusion obtained by Mas et al. that associativity and bisymmetry are equivalent for commutative smooth aggregation operators on a finite chain.

Morphisms on $${ EMV}$$ EMV -algebras and their applications

For a new class of algebras, called \({ EMV}\)-algebras, every idempotent element a determines an \({ MV}\)-algebra which is important for the structure of the \({ EMV}\)-algebra. Therefore, instead of standard homomorphisms of \({ EMV}\)-algebras, we introduce \({ EMV}\)-morphisms as a family of \({ MV}\)-homomorphisms from \({ MV}\)-algebras [0, a] into other ones. \({ EMV}\)-morphisms enable us to study categories of \({ EMV}\)-algebras where objects are \({ EMV}\)-algebras and morphisms are special classes of \({ EMV}\)-morphisms. The category is closed under product. In addition, we define free \({ EMV}\)-algebras on a set X with respect to \({ EMV}\)-morphisms. If X is finite, then a free \({ EMV}\)-algebra on X is termwise equivalent to the free \({ MV}\)-algebra on X. For an infinite set X, the same is true introducing a so-called weakly free \({ EMV}\)-algebra.

Petri nets are dioids: a new algebraic foundation for non-deterministic net theory

In a seminal paper Montanari and Meseguer have shown that an algebraic interpretation of Petri nets in terms of commutative monoids can be used to provide an elegant characterisation of the deterministic computations of a net, accounting for their sequential and parallel composition. A smoother and more complete theory for deterministic computations has been later developed by relying on the concept of pre-net, a variation of Petri nets with a non-commutative flavor. This paper shows that, along the same lines, by adding an (idempotent) operation and thus considering dioids (idempotent semirings) rather than just monoids, one can faithfully characterise the non-deterministic computations of a net.

π-Baer rings

We say a ring [Formula: see text] is [Formula: see text]-Baer if the right annihilator of every projection invariant left ideal of [Formula: see text] is generated by an idempotent element of [Formula: see text]. In this paper, we study connections between the [Formula: see text]-Baer condition and related conditions such as the Baer, quasi-Baer and [Formula: see text]-extending conditions. The [Formula: see text]-by-[Formula: see text] generalized triangular and the [Formula: see text]-by-[Formula: see text] triangular [Formula: see text]-Baer matrix rings are characterized. Also, we prove that a [Formula: see text]-by-[Formula: see text] full matrix ring over a [Formula: see text]-Baer ring is a [Formula: see text]-Baer ring. In contrast to the Baer condition, it is shown that the [Formula: see text]-Baer condition transfers from a base ring to many of its polynomial extensions. Examples are provided to illustrate and delimit our results.

Orbit Codes over M2(ℱq) and Their Homogeneous Distance

Let q be a power of a prime. The lattice of one-sided ideals of the finite unital non-commutative Frobenius ring M2(ℱq) of 2 × 2 matrices over the Galois field (ℱq) is completely analyzed. It turns out that M2(ℱq) is a principal left semi-local ring in which each left ideal is generated by an idempotent element. The explicit forms of the non-trivial idempotents of M2(ℱq) are determined to give q + 1 proper non-trivial left maximal ideals each with q elements. These are exactly the minimal left ideals as well. Using the structure of M2(ℱq) as a partial ordering of ideals, the generalized Möbius and Euler phi functions are applied to derive the explicit form of the homogeneous weight function on M2(ℱq). This weight depends on whether the element is the zero element, a zero divisor or a unit. A zero divisor gives the largest homogeneous weight. Moreover, orbit codes over M2(ℱq) are constructed via the action of the general linear group GL(2, q) on M2(ℱq) by left translation. The orbit determined by a nonzero nonunit idempotent element of M2(ℱq) forms the nonzero elements of a minimal left ideal of M2(ℱq) which are all zero divisors. Consequently, it is shown that the minimum homogeneous distance of the orbit code generated by a nonzero nonunit idempotent element of M2(ℱq) approaches the Plotkin upper bound as the field size q becomes larger. Analogous results are obtained when the lattice of right ideals is considered and the action of GL(2, q) on M2(ℱq) by right translation is used instead.

Idempotent conjunctive and disjunctive combination of belief functions by distance minimization

Idempotence is a desirable property when cautiousness is wanted in an information fusion process, since in this case combining identical information should not lead to the reinforcement of some hypothesis. Idempotent operators also guarantee that identical information items are not counted twice in the fusion process, a very important property in decentralized applications where the information origin cannot always be tracked (ad-hoc wireless networks are typical examples). In the theory of belief functions, a sound way to combine conjunctively multiple information items is to design a combination rule that selects the least informative element among a subset of belief functions more informative than each of the combined ones. In contrast, disjunctive rules can be retrieved by selecting the most informative element among a subset of belief functions less informative than each of the combined ones. One interest of such approaches is that they provide idempotent rules by construction.The notions of less and more informative are often formalized through partial orderings extending usual set-inclusion, yet the only two informative partial orders that provide a straightforward idempotent rule leading to a unique result are those based on the conjunctive and disjunctive weight functions. In this article, we show that other partial orders can achieve a similar goal when the problem is slightly relaxed into a distance optimization one. Building upon previous work, this paper investigates the use of distances compatible with informative partial orders to determine a unique solution to the combination problem. The obtained operators are conjunctive/disjunctive, idempotent and commutative, but lack associativity. They are, however, quasi-associative allowing sequential combinations at no extra complexity. Some experiments demonstrate interesting discrepancies as compared to existing approaches, notably with the aforementioned rules relying on weight functions.

Some types of ring elements in Ore extensions over noncommutative rings

Let [Formula: see text] be an associative unital ring with an endomorphism [Formula: see text] and [Formula: see text]-derivation [Formula: see text]. Some types of ring elements such as the units and the idempotents play distinguished roles in noncommutative ring theory, and will play a central role in this work. In fact, we are interested to study the unit elements, the idempotent elements, the von Neumann regular elements, the [Formula: see text]-regular elements and also the von Neumann local elements of the Ore extension ring [Formula: see text], when the base ring [Formula: see text] is a right duo ring which is [Formula: see text]-compatible. As an application, we completely characterize the clean elements of the Ore extension ring [Formula: see text], when the base ring [Formula: see text] is a right duo ring which is [Formula: see text]-compatible.

Properties of non-associative MV-algebras

Abstract Non-associative MV-algebras (NMV-algebras) (A, ⊕, ¬, 0) were introduced in [CHAJDA, I.—KÜHR, J.: A non-associative generalization of MV-algebras, Math. Slovaca 57 (2007), 301–312]. In the present paper we prove some properties of these algebras, investigate when intervals of the form [a, 1] can be made into NMV-algebras in some natural way and consider idempotent elements and derivations of NMV-algebras. Moreover, we study decompositions of NMV-algebras and characterize the congruences on NMV-algebras by means of so-called filters.

Systematic Constructions of Rotation Symmetric Bent Functions, 2-Rotation Symmetric Bent Functions, and Bent Idempotent Functions

Rotation symmetric bent functions and their generation two-rotation symmetric bent functions are two classes of cryptographically significant Boolean functions. However, few constructions have been presented in the literature, which either have restriction on integer $n$ or have algebraic degree no more than 4. In this paper, for any even integer $n\ge 4$ , three classes of bent functions are presented respectively. Most notably, the proposed $n$ -variable rotation symmetric bent functions and two-rotation symmetric bent functions can have any possible algebraic degree ranging from 2 to $n/2$ . Besides, we obtain bent idempotent functions with the maximal algebraic degree $n/2$ .

Level sets of $$(p,e-p)$$(p,e-p) outer generalized pseudo spectrum

Let A be a complex Banach algebra with unit e. Let p be a non trivial idempotent element in A and e>0. For a∈A, it is proved that the interior of the level set of (p,e−p)−e pseudo spectrum of a is empty in the unbounded component of (p,e−p) resolvent set of a. An example is constructed to show that the condition ‘unbounded component’ can not be dropped. Further, it is proved this ‘unbounded component’ can be dropped in the case when A is B(X) where X is a complex uniformly convex Banach space. That is, if T∈B(X) then interior of the level set of (p,I−p)−e pseudo spectrum is empty in (p,I−p) resolvent set of T.

A characterization of n-associative, monotone, idempotent functions on an interval that have neutral elements

We investigate monotone idempotent n-ary semigroups and provide a generalization of the Czogala–Drewniak Theorem, which describes the idempotent monotone associative functions having a neutral element. We also present a complete characterization of idempotent monotone n-associative functions on an interval that have neutral elements.

π-Clean and strongly π-clean rings

In this paper, π-regular (resp. strongly π-regular) and idempotent elements in a ring is used to define π-clean (resp. strongly π-clean) rings. A ring is called π-clean (resp. strongly π-clean) if each of its elements is the sum of a π-regular (resp. strongly π-regular) and an idempotent element. The purpose of this research is to give some results on π-clean and strongly π-clean rings and study π-clean and strongly π-clean group rings. Also, this paper is involved with investigating F2C2 as a social group and measuring influence a member of it’s rather than others.

Convolution lattices

We propose two convolution operations on the set of functions between two bounded lattices and investigate the algebraic structure they constitute, in particular the lattice laws they satisfy. Each of these laws requires the restriction to a specific subset of functions, such as normal, idempotent or convex functions. Combining all individual results, we identify the maximal subsets of functions resulting in a bounded lattice, and show this result to be equivalent to the distributivity of the lattice acting as domain of the functions. Furthermore, these lattices turn out to be distributive as well. Additionally, we show that for the larger subset of idempotent functions, although not satisfying the absorption laws, the convolution operations satisfy the Birkhoff equation.

On internally cancellable rings

The concept of internally cancellable rings has been extensively studied in the literature. This paper seeks to continue the study of these rings and find some new characterizations. It is proved that [Formula: see text] is “IC”, if and only if for each regular element [Formula: see text], and idempotent element [Formula: see text] with [Formula: see text], there exists [Formula: see text] such that [Formula: see text] is a unit (alternatively, unit-regular element) in [Formula: see text] and [Formula: see text]. In case the ring [Formula: see text] has the summand sum property, we indicate that [Formula: see text] is IC, if and only if for each regular element [Formula: see text], and element [Formula: see text] with [Formula: see text], there exists an idempotent [Formula: see text], such that [Formula: see text] is a unit in [Formula: see text] and [Formula: see text].

Inner actions of weak Hopf algebras

Let [Formula: see text] be an associative ring and [Formula: see text] idempotent elements of [Formula: see text]. In this paper we introduce the notion of [Formula: see text]-invertibility for an element of [Formula: see text] and use it to define inner actions of weak Hopf algebras. Given a weak Hopf algebra [Formula: see text] and an algebra [Formula: see text], we present sufficient conditions for [Formula: see text] to admit an inner action of [Formula: see text]. We also prove that if [Formula: see text] is a left [Formula: see text]-module algebra then [Formula: see text] acts innerly on the smash product [Formula: see text] if and only if [Formula: see text] is a quantum commutative weak Hopf algebra.