Idempotency Research Articles

Representations of power-associative train algebras of rank 4

This paper is devoted to the study of four-dimensional commutative power-associative non-Jordan train algebras. The existence of a unique orthogonal idempotent element in such kind of algebras enables us to define a non-zero algebra homomorphism over their ground field, which gives rise to train algebras of rank 4. We describe the irreducible representations over an algebraically closed field of characteristic prime to 30 by means of multilinear identities which define it. Furthermore, we provide sufficient conditions to guarantee whenever a representation is decomposable.

Exceptional sequences and idempotent functions

We prove that there is a one-to-one correspondence between the following three sets: idempotent functions on a set of size n, complete exceptional sequences of linear radical square zero Nakayama algebras of rank n and rooted labeled forests with n nodes and height of at most one. Therefore, the number of exceptional sequences is given by the sum ∑j=1nnjjn−j.

On generalized ∗-reverse derivable maps

Let R be a ring with involution containing a nontrivial symmetric idempotent element e and δ: R → R be a generalized ∗-reverse derivable map. In this paper, our aim is to show that under some suitable restrictions imposed on R every generalized ∗-reverse derivable map of R is additive.

A semigroup of theories and its lattice of idempotent elements

A semigroup of theories and its lattice of idempotent elements

Special idempotents and projections

We define, for any special matching of a finite graded poset, an idempotent, regressive and order preserving function. We consider the monoid generated by such functions. We call special idempotent any idempotent element of this monoid. They are interval retracts. Some of them realize a kind of parabolic map and are called special projections. We prove that, in Eulerian posets, the image of a special projection, and its complement, are graded induced subposets. In a finite Coxeter group, all projections on right and left parabolic quotients are special projections, and some projections on double quotients too. We extend our results to special partial matchings.

Semigroups of an inductive composition of terms

The set of all [Formula: see text]-ary terms of type [Formula: see text] together with a binary operation derived from a superposition [Formula: see text] forms various forms of semigroups. One may generalize such binary operation by deriving it from an inductive composition of terms and call it an inductive product. However, this operation is not associative on the same base set but it becomes associative when all elements of subterms of a fixed term used in an inductive product except itself are excluded from the base set. Hence, a semigroup is formed. In this paper, we mainly focus on the algebraic structures of this semigroup such as idempotent elements, elements associating with each type of regularity condition, and Green’s relations. The formulae of complexity of inducted terms are also under investigation.

Idempotent, model, and Toeplitz operators attaining their norms

We study idempotent, model, and Toeplitz operators that attain the norm. Notably, we prove that if Q is a backward shift invariant subspace of the Hardy space H2(D), then the model operator SQ attains its norm. Here SQ=PQMz|Q, the compression of the shift Mz on the Hardy space H2(D) to Q.

Quasi-uniform structures determined by closure operators

We demonstrate a one-to-one correspondence between idempotent closure operators and the so-called saturated quasi-uniform structures on a category C. Not only this result allows to obtain a categorical counterpart P of the Császár-Pervin quasi-uniformity P, that we characterize as a transitive quasi-uniformity compatible with an idempotent interior operator, but also permits to describe those topogenous orders that are induced by a transitive quasi-uniformity on C. The categorical counterpart P⁎ of P−1 is characterized as a transitive quasi-uniformity compatible with an idempotent closure operator. When applied to other categories outside topology P allows, among other things, to generate a family of idempotent closure operators on Grp, the category of groups and group homomorphisms, determined by the normal closure.

Some remarks on evolution algebras corresponding to permutations

In this paper, we describe absolute nilpotent and some idempotent elements of an [Formula: see text]-dimensional evolution algebra corresponding to two permutations and we decompose such algebras to the direct sum of evolution algebras corresponding to cycles of the permutations.

$$L^p$$-projections on subspaces and quotients of Banach spaces

The aim of this paper is to study \(L^p\)-projections, a notion introduced by Cunningham in 1953, on subspaces and quotients of complex Banach spaces. An \(L^p\)-projection on a Banach space X, for \(1\le p \le +\infty\), is an idempotent operator P satisfying \(\Vert f\Vert _X = \Vert ( \Vert P(f)\Vert _X, \Vert (I-P)(f)\Vert _X) \Vert _{\ell _{p}}\) for all \(f \in X\). This is an \(L^p\) version of the equality \(\Vert f\Vert ^2=\Vert Q(f)\Vert ^2 + \Vert (I-Q)(f)\Vert ^2\), valid for orthogonal projections on Hilbert spaces. We study the relationships between \(L^p\)-projections on a Banach space X and those on a subspace F, as well as relationships between \(L^p\)-projections on X and those on the quotient space X/F. All the results in this paper are true for \(1<p<+\infty\), \(p\ne 2\). The cases \(p=1,2\) or \(+\infty\) can exhibit different behavior. In this regard, we give a complete description of \(L^{\infty }\)-projections on spaces \(L^{\infty }(\Omega )\). For this, we introduce a notion of p-orthogonality for two elements x, y by requiring that \(\text {Span}(x,y)\) admits an \(L^p\)-projection separating x and y. We also introduce the notion of maximal \(L^p\)-projections for X, that is \(L^p\)-projections defined on a subspace G of X that cannot be extended to \(L^p\)-projections on larger subspaces. We prove results concerning \(L^p\)-projections and p-orthogonality of general Banach spaces or on Banach spaces with additional properties. Generalizations of some results to spaces \(L^p(\Omega ,X)\) as well as some results about \(L^q\)-projections on subspaces of \(L^p(\Omega )\) are also discussed.

On the idempotent and nilpotent sum numbers of matrices over certain indecomposable rings and related concepts

We investigate a few special decompositions in arbitrary rings and matrix rings over indecomposable rings into nilpotent and idempotent elements. Moreover, we also define and study the nilpotent sum trace number of nilpotent matrices over an arbitrary ring. Some related notions are explored as well.

Entropy Based Deep Attention Mechanism (EDAM) To Mitigate Denial of Service (Dos) Attack Orchestrated Through Idempotent Operation

Entropy Based Deep Attention Mechanism (EDAM) To Mitigate Denial of Service (Dos) Attack Orchestrated Through Idempotent Operation

A Semigroup of Theories and Its Lattice of Idempotent Elements

A Semigroup of Theories and Its Lattice of Idempotent Elements

The star order for idempotent operators

The purpose of this note is to consider the lattice properties of (the set of all idempotent operators on a Hilbert space ) with respect to the star partial order. In the domain of we prove that the star infimum always exists for an arbitrary nonempty subset of . Also, we present the necessary and sufficient conditions for the existence of the star supremum for an arbitrary nonempty subset of . In particular, an explicit representation of the star supremum is established for two arbitrary idempotent operators in .

Cartesian closedness in categories with an idempotent closure operator and closed morphisms

Given a subobject-structured category $$\mathcal X$$ , we construct a new category whose objects are the pairs (X, c) where X is an $$\mathcal X$$ -object and c is an idempotent, monotonic and extensive endomap of the subobject lattice of X, and whose morphisms between objects are the closed maps between the corresponding subobject lattices. We give a sufficient condition on $$\mathcal X$$ for the new category to be cartesian closed.

On the Jacobson radical of a groupoid graded ring

Let R be a ring graded by a finite cancellative partial groupoid S, and let E(S) denote the set of all idempotent elements of S. The Jacobson radical of a ring A is denoted by J(A), and P(A) denotes the prime radical of A. In this paper we affirmatively answer two questions posed by Andrei Kelarev. In particular we prove that: i) J(R)∩Re=J(Re) for every e∈E(S); ii) J(R)n is contained in the largest homogeneous quasi-regular ideal of R for some integer n. We moreover prove that P(R)∩Re=P(Re) for every e∈E(S), and investigate the questions of nilness and nilpotency of the Jacobson radical J(R).

FUNCTORS AND SPACES IN IDEMPOTENT MATHEMATICS

Idempotent mathematics is a branch of mathematics in which idempotent operations (for example, max) on the set of reals play a central role. In recent decades, we have seen intensive research in this direction. The principle of correspondence (this is an informal principle analogous to the Bohr correspondence principle in the quantum mechanics) asserts that each meaningful concept or result of traditional mathematics corresponds to a meaningful concept or result of idempotent mathematics. In particular, to the notion of probability measure there corresponds that if Maslov measure (also called idempotent measure) as well as more recent notion of max-min measure. Also, there are idempotent counterparts of the convex sets; these include the so-called max-plus and max min convex sets. Methods of idempotent mathematics are used in optimization problems, dynamic programming, mathematical economics, game theory, mathematical biology and other disciplines. In this paper we provide a survey of results that concern algebraic and geometric properties of the functors of idempotent and max-min measures.

On general closure operators and quasi factorization structures

In this article the notions of quasi mono (epi) as a generalization of mono (epi), (quasi weakly hereditary) general closure operator $mathbf{C}$ on a category $mathcal{X}$ with respect to a class $mathcal{M}$ of morphisms, and quasi factorization structures in a category $mathcal{X}$ are introduced. It is shown that under certain conditions, if $(mathcal{E}, mathcal{M})$ is a quasi factorization structure in $mathcal{X}$, then $mathcal{X}$ has a quasi right $mathcal{M}$-factorization structure and a quasi left $mathcal{E}$-factorization structure. It is also shown that for a quasi weakly hereditary and quasi idempotent QCD-closure operator with respect to a certain class $mathcal{M}$, every quasi factorization structure $(mathcal{E}, mathcal{M})$ yields a quasi factorization structure relative to the given closure operator; and that for a closure operator with respect to a certain class $mathcal{M}$, if the pair of classes of quasi dense and quasi closed morphisms forms a quasi factorization structure, then the closure operator is both quasi weakly hereditary and quasi idempotent. Several illustrative examples are provided.

Evolution algebras and dynamical systems of a worm propagation model

ABSTRACT We consider an evolution algebra identifying the coefficients of SIS–SIR worm propagation models as the structure constants of the algebra. The basic properties of this algebra are studied. We prove that it is a commutative (and hence flexible), not associative and baric algebra. We describe the full set of idempotent elements and the full set of absolute nilpotent elements. We find all the fixed points of the dynamical systems. We also study several properties of the algebra connecting them to dynamical systems.

On the Auslander algebra of one commutative semigroup of finite representation type

We consider matrix representations of standard oversemigroups of the semigroup generated by two mutually annihilating idempotent elements. For the only such over­semigroup of finite representation type their algebra Auslander is described. Cite as: O. V. Zubaruk, “On the Auslander algebra of one commutative semigroup of finite representation type,” Prykl. Probl. Mekh. Mat. , Issue 18, 43–47 (2020) (in Ukrainian), https://doi.org/10.15407/apmm2020.18.43-47