Showing papers on "Idempotence published in 2018"
TL;DR: It is shown that idempotent uninorms on an arbitrary bounded lattice need not always be internal (with the extended definition of the term ”internal”).
33 citations
TL;DR: It is proved that an idempotent nullnorm may not always exist on an arbitrary bounded lattice, and a construction method is proposed to obtain idem Potentnullnorms on a bounded lattices with an additional constraint on a for the given zero element.
33 citations
TL;DR: In this paper, an axiomatic characterization of idempotent discrete uninorms by means of three conditions only: conservativeness, symmetry, and non-decreasing monotonicity is provided.
30 citations
TL;DR: In this article, it was shown that for every n × n matrix A over the field Z 2 there exists an idempotent matrix E such that (A − E ) 4 = 0.
30 citations
TL;DR: In this article, the authors describe a method for constructing a weight structure on a triangulated category such that any bounded above or below the weight structure extends to any idempotent extension of the category.
Abstract: We describe a new method for constructing a weight structure $w$ on a triangulated category $C$.
For a given $C$ and $w$ it allow us to give a fairly comprehensive (and new) description of those triangulated categories consisting of retracts of objects of $C$ (i.e., of subcategories of the Karoubi envelope of $C$ that contain $C$; we call them idempotent extensions of $C$) such that $w$ extends to them. In particular, any bounded above or below $w$ extends to any idempotent extension of $C$. We also discuss the applications of our results to certain triangulated categories of (\"relative\") motives.
28 citations
TL;DR: In this article, the dominant dimensions of Nakayama algebras and more general algesbras A with an idempotent e such that there is a minimal faithful injective-projective module eA and such that eAe is a Nakaya algebra are given.
26 citations
TL;DR: In this article, an idempotent analogue of the exterior algebra for the theory of tropical linear spaces (and valuated matroids) can be seen in close analogy with the classical Grassmann algebra formalism for linear spaces, and it is shown that an arbitrary d-tensor satisfies the tropical Plucker relations (valuated exchange axiom) if and only if the dth wedge power of the kernel of wedge multiplication is free of rank one.
Abstract: We introduce an idempotent analogue of the exterior algebra for which the theory of tropical linear spaces (and valuated matroids) can be seen in close analogy with the classical Grassmann algebra formalism for linear spaces. The top wedge power of a tropical linear space is its Plucker vector, which we view as a tensor, and a tropical linear space is recovered from its Plucker vector as the kernel of the corresponding wedge multiplication map. We prove that an arbitrary d-tensor satisfies the tropical Plucker relations (valuated exchange axiom) if and only if the dth wedge power of the kernel of wedge-multiplication is free of rank one. This provides a new cryptomorphism for valuated matroids, including ordinary matroids as a special case.
24 citations
TL;DR: In this article, the authors completely determine the rings for which every element is a sum of a nilpotent, an idempotent and a tripotent that commute with one another.
Abstract: We completely determine the rings for which every element is a sum of a nilpotent, an idempotent and a tripotent that commute with one another, and the rings for which every element is a sum of a nilpotent and two tripotents that commute with one another.
23 citations
27 Feb 2018
TL;DR: In this article, it was shown that quantum teleportation is only successfully supported on the intersection of Alice and Bob's causal future, and that relativistic quantum information theory using methods entirely internal to monoidal categories.
Abstract: The category of Hilbert modules may be interpreted as a naive quantum field theory over a base space. Open subsets of the base space are recovered as idempotent subunits, which form a meet-semilattice in any firm braided monoidal category. There is an operation of restriction to an idempotent subunit: it is a graded monad on the category, and has the universal property of algebraic localisation. Spacetime structure on the base space induces a closure operator on the idempotent subunits. Restriction is then interpreted as spacetime propagation. This lets us study relativistic quantum information theory using methods entirely internal to monoidal categories. As a proof of concept, we show that quantum teleportation is only successfully supported on the intersection of Alice and Bob's causal future.
18 citations
Posted Content•
TL;DR: In this paper, it was shown that a set-theoretic solution of the Yang-Baxter equation on a finite set can be degenerate or even idempotent.
Abstract: Let $r:X^{2}\rightarrow X^{2}$ be a set-theoretic solution of the Yang-Baxter equation on a finite set $X$. It was proven by Gateva-Ivanova and Van den Bergh that if $r$ is non-degenerate and involutive then the algebra $K\langle x \in X \mid xy =uv \mbox{ if } r(x,y)=(u,v)\rangle$ shares many properties with commutative polynomial algebras in finitely many variables; in particular this algebra is Noetherian, satisfies a polynomial identity and has Gelfand-Kirillov dimension a positive integer. Lebed and Vendramin recently extended this result to arbitrary non-degenerate bijective solutions. Such solutions are naturally associated to finite skew left braces. In this paper we will prove an analogue result for arbitrary solutions $r_B$ that are associated to a left semi-brace $B$; such solutions can be degenerate or can even be idempotent. In order to do so we first describe such semi-braces and we prove some decompositions results extending results of Catino, Colazzo, and Stefanelli.
18 citations
TL;DR: In this paper, the authors investigate the structure of the twisted Brauer monoid and give necessary and sufficient conditions for an ideal to be idempotent generated, and obtain formulae for the rank (smallest size of a generating set) of each principal ideal.
Abstract: We investigate the structure of the twisted Brauer monoid , comparing and contrasting it with the structure of the (untwisted) Brauer monoid . We characterize Green's relations and pre-orders on , describe the lattice of ideals and give necessary and sufficient conditions for an ideal to be idempotent generated. We obtain formulae for the rank (smallest size of a generating set) and (where applicable) the idempotent rank (smallest size of an idempotent generating set) of each principal ideal; in particular, when an ideal is idempotent generated, its rank and idempotent rank are equal. As an application of our results, we describe the idempotent generated subsemigroup of (which is not an ideal), as well as the singular ideal of (which is neither principal nor idempotent generated), and we deduce that the singular part of the Brauer monoid is idempotent generated, a result previously proved by Maltcev and Mazorchuk.
TL;DR: In this article, it was shown that an idempotent a in the Griess algebra is indecomposable if and only if its Peirce 1-eigenspace is one-dimensional.
TL;DR: In this paper, the authors consider the preservation of properties of being finitely generated, being residually finite under direct products in the context of different types of algebraic structures and identify as broad classes as possible in which the expected preservation results (A × B A × B satisfies property P P if and only if A and B satisfy P P ) may fail outside those classes.
TL;DR: In this article, the authors studied the spectral properties of nonassociative algebras from the idempotent point of view and showed that there are at least n-1 nontrivial obstructions on the Peirce spectrum of a generic NA algebra of dimension n.
Abstract: In this paper, we study the variety of all nonassociative (NA) algebras from the idempotent point of view. We are interested, in particular, in the spectral properties of idempotents when algebra is generic, i.e. idempotents are in general position. Our main result states that in this case, there exist at least $n-1$ nontrivial obstructions (syzygies) on the Peirce spectrum of a generic NA algebra of dimension $n$. We also discuss the exceptionality of the eigenvalue $\lambda=\frac12$ which appears in the spectrum of idempotents in many classical examples of NA algebras and characterize its extremal properties in metrised algebras.
TL;DR: In this article, a one-to-one correspondence between shifts of group-like projections on a locally compact quantum group and contractive idempotent functionals on the dual quantum group was shown.
Abstract: A one-to-one correspondence between shifts of group-like projections on a locally compact quantum group 𝔾 which are preserved by the scaling group and contractive idempotent functionals on the dual...
TL;DR: In this paper, a generalization of the Czogala-Drewniak Theorem for monotone idempotent n-ary semigroups is presented.
Abstract: 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.
TL;DR: In this paper, it was shown that the differential polynomial ring R[X; δ] cannot be mapped onto a ring with a non-zero idempotent ring.
Abstract: Let δ be a derivation of a locally nilpotent ring R. Then the differential polynomial ring R[X; δ] cannot be mapped onto a ring with a non-zero idempotent. This answers a recent question by Greenfeld, Smoktunowicz and Ziembowski.
TL;DR: The aim of this paper is to present a generating set for the class of intermediate (or, equivalently, idempotent) aggregation functions, which consists of lattice operations and certain ternary idem Potent aggregation functions.
TL;DR: In this article, Krasnov and Tkachev characterized the combinatorial stratification of the variety of real generic algebras and showed that there exist exactly three different homotopic types of such algebraic types and related this result to potential applications and known facts from qualitative theory of quadratic ODEs.
Abstract: Using the syzygy method, established in our earlier paper (Krasnov and Tkachev, Honor of Wolfgang SprsigTrends Math, Birkhauser/Springer Basel AG, Basel, 2018), we characterize the combinatorial stratification of the variety of two-dimensional real generic algebras. We show that there exist exactly three different homotopic types of such algebras and relate this result to potential applications and known facts from qualitative theory of quadratic ODEs. The genericity condition is crucial. For example, the idempotent geometry in Clifford algebras or Jordan algebras of Clifford type is very different: such algebras always contain nontrivial submanifolds of idempotents.
TL;DR: The notion of invariant idempotent measures in complete metric spaces was introduced in this article, where the authors proved the existence and uniqueness of the notion of in-homogeneous invariant probability measures for an iterated function system in a complete metric space.
Abstract: The idempotent mathematics is a part of mathematics in which arithmetic operations in the reals are replaced by idempotent operations. In the idempotent mathematics, the notion of idempotent measure (Maslov measure) is a counterpart of the notion of probability measure. The idempotent measures found numerous applications in mathematics and related areas, in particular, the optimization theory, mathematical morphology, and game theory.
In this note we introduce the notion of invariant idempotent measure for an iterated function system in a complete metric space. This is an idempotent counterpart of the notion of invariant probability measure defined by Hutchinson. Remark that the notion of invariant idempotent measure was previously considered by the authors for the class of ultrametric spaces.
One of the main results is the existence and uniqueness theorem for the invariant idempotent measures in complete metric spaces. Unlikely to the corresponding Hutchinson's result for invariant probability measures, our proof does not rely on metrization of the space of idempotent measures.
An analogous result can be also proved for the so-called in-homogeneous idempotent measures in complete metric spaces.
Also, our considerations can be extended to the case of the max-min measures in complete metric spaces.
TL;DR: Proving that for each fixed width m there is a weakest loop condition (that is, one entailed by all others), and obtaining a new and short proof of the recent celebrated result stating that there exists a concrete loop condition of width 3 which is entailed in any non‐trivial idempotent, possibly infinite, algebra.
Abstract: We initiate the systematic study of loop conditions of arbitrary finite width. Each loop condition is a finite set of identities of a particular shape, and satisfaction of these identities in an algebra is characterized by it forcing a constant tuple into certain invariant relations on powers of the algebra.
By showing the equivalence of various loop conditions, we are able to provide a new and short proof of the recent celebrated result stating the existence of a weakest non-trivial idempotent strong Mal'cev condition.
We then consider pseudo-loop conditions, a modification suitable for oligomorphic algebras, and show the equivalence of various pseudo-loop conditions within this context. This allows us to provide a new and short proof of the fact that the satisfaction of non-trivial identities of height 1 in a closed oligomorphic core implies the satisfaction of a fixed single identity.
TL;DR: Given an idempotent element e of a commutative ring R and a vertex-set V(Γe(R)) of R associated with e, define the graph Γe (R) associated with the element e to be the (undirected) graph with vertex set V( Γ e(R) = {a∈R | there exists b...
Abstract: Given 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...
Posted Content•
TL;DR: In this article, the authors investigate under which conditions the space of idempotent measures is an absolute retract and the ideme-potent barycenter map is soft.
Abstract: We investigate under which conditions the space of idempotent measures is an absolute retract and the idempotent barycenter map is soft.
Posted Content•
TL;DR: In this article, the authors studied lattice operations on the set of idempotent states on a locally compact quantum group corresponding to the operations of intersection of compact subgroups and forming the subgroup generated by two compact subgroup Normal (i.e., weakly continuous) states are investigated.
Abstract: We study lattice operations on the set of idempotent states on a locally compact quantum group corresponding to the operations of intersection of compact subgroups and forming the subgroup generated by two compact subgroups Normal ($\sigma$-weakly continuous) idempotent states are investigated and a duality between normal idempotent states on a locally compact quantum group $\mathbb{G}$ and on its dual $\widehat{\mathbb{G}}$ is established Additionally we analyze the question when a left coideal corresponding canonically to an idempotent state is finite dimensional and give a characterization of normal idempotent states on compact quantum groups
TL;DR: In this paper, the authors studied the L1(M, τ) space of integrable operators affiliated to the von Neumann algebra M = B(H) of all bounded linear operators on H which are endowed with the canonical trace τ = tr.
Abstract: Suppose that M is a von Neumann algebra of operators on a Hilbert space H and τ is a faithful normal semifinite trace on M. Let E, F and G be ideal spaces on (M, τ). We find when a τ-measurable operator X belongs to E in terms of the idempotent P of M. The sets E+F and E·F are also ideal spaces on (M, τ); moreover, E·F = F·E and (E+F)·G = E·G+F·G. The structure of ideal spaces is modular. We establish some new properties of the L1(M, τ) space of integrable operators affiliated to the algebra M. The results are new even for the *-algebra M = B(H) of all bounded linear operators on H which is endowed with the canonical trace τ = tr.
TL;DR: An essential improvement of the result above is presented by presenting a new generating set of the clone of idempotent aggregation functions on bounded lattices, where a bit artificial ternary functions are substituted here by natural (binary) lattice a-medians and certain binary characteristic functions.
Posted Content•
TL;DR: In this article, a criterion for a semigroup identity to hold in the monoid of upper unitriangular matrices with entries in a commutative semiring was established.
Abstract: We establish a criterion for a semigroup identity to hold in the monoid of $n \times n$ upper unitriangular matrices with entries in a commutative semiring $S$. This criterion is combinatorial modulo the arithmetic of the multiplicative identity element of $S$. In the case where $S$ is idempotent, the generated variety is the variety $\mathbf{J_{n-1}}$, which by a result of Volkov is generated by any one of: the monoid of unitriangular Boolean matrices, the monoid $R_n$ of all reflexive relations on an $n$ element set, or the Catalan monoid $C_n$. We propose $S$-matrix analogues of these latter two monoids in the case where $S$ is an idempotent semiring whose multiplicative identity element is the `top' element with respect to the natural partial order on $S$, and show that each generates $\mathbf{J_{n-1}}$. As a consequence we obtain a complete solution to the finite basis problem for lossy gossip monoids.
TL;DR: This work reports on progress in characterizing K -valued FCA in algebraic terms, and states the importance of FCA-related concepts for dual order homomorphisms of linear spaces over idempotent semifields, specially congruences, the lattices of extents, intents and formal concepts.
01 Mar 2018
TL;DR: It is shown that if an idempotent semiring is equipped with an involution which satisfies certain conditions, then it can be organized into a residuated lattice satisfying the double negation law.
Abstract: Every residuated lattice can be considered as an idempotent semiring. Conversely, if an idempotent semiring is finite, then it can be organized into a residuated lattice. Unfortunately, this does not hold in general. We show that if an idempotent semiring is equipped with an involution which satisfies certain conditions, then it can be organized into a residuated lattice satisfying the double negation law. Also conversely, every residuated lattice satisfying the double negation law can be considered as an idempotent semiring with an involution satisfying the mentioned conditions.
TL;DR: It is proved that pseudovarieties of D-monoids bijectively correspond to varieties of regular languages in C, which generalizes Eilenberg’s concept of a variety of languages, which corresponds to choosing as C the category of Boolean algebras.
Abstract: For finite automata as coalgebras in a category C, we study languages they accept and varieties of such languages. This generalizes Eilenberg’s concept of a variety of languages, which corresponds to choosing as C the category of Boolean algebras. Eilenberg established a bijective correspondence between pseudovarieties of monoids and varieties of regular languages. In our generalization, we work with a pair C/D of locally finite varieties of algebras that are predual, i.e., dualize on the level of finite algebras, and we prove that pseudovarieties of D-monoids bijectively correspond to varieties of regular languages in C. As one instance, Eilenberg’s result is recovered by choosing D = sets and C = Boolean algebras. Another instance, Pin’s result on pseudovarieties of ordered monoids, is covered by taking D = posets and C = distributive lattices. By choosing as C amp;equals; D the self-predual category of join-semilattices, we obtain Polak’s result on pseudovarieties of idempotent semirings. Similarly, using the self-preduality of vector spaces over a finite field K, our result covers that of Reutenauer on pseudovarieties of K-algebras. Several new variants of Eilenberg’s theorem arise by taking other predualities, e.g., between the categories of non-unital Boolean rings and of pointed sets. In each of these cases, we also prove a local variant of the bijection, where a fixed alphabet is assumed and one considers local varieties of regular languages over that alphabet in the category C.