Showing papers on "Idempotence published in 2016"
TL;DR: This work shows the existence of idempotent uninorms on L for any element e ź L\{0, 1} playing the role of a neutral element, and studies and proposes some new construction methods to obtain uninormS on bounded lattices.
83 citations
TL;DR: In this paper, it was shown that the endomorphism algebra of a corresponding Cohen-Macaulay module T over the algebra B used to categorise the cluster structure of Gr(k, n) by Jensen-King-Su can be realized as the boundary algebra of A, that is, the subalgebra eAe for an idempotent e corresponding to the boundary.
Abstract: We associate a dimer algebra A to a Postnikov diagram D (in a disk) corresponding to a cluster of minors in the cluster structure of the Grassmannian Gr(k, n). We show that A is isomorphic to the endomorphism algebra of a corresponding Cohen-Macaulay module T over the algebra B used to categorify the cluster structure of Gr(k, n) by Jensen-King-Su. It follows that B can be realised as the boundary algebra of A, that is, the subalgebra eAe for an idempotent e corresponding to the boundary of the disk. The construction and proof uses an interpretation of the diagram D, with its associated plabic graph and dual quiver (with faces), as a dimer model with boundary. We also discuss the general surface case, in particular computing boundary algebras associated to the annulus.
83 citations
TL;DR: In this article, the authors proved that every element of a ring is a sum of an idempotent and a tripotent that commute if and only if the identity of the tripotent is known.
Abstract: Let be a ring. The following results are proved. Every element of is a sum of an idempotent and a tripotent that commute if and only if has the identity if and only if , where is Boolean with a group of exponent and is zero or a subdirect product of . Every element of is either a sum or a difference of two commuting idempotents if and only if , where is Boolean with or and is zero or a subdirect product of . Every element of is a sum of two commuting tripotents if and only if , where is Boolean with a group of exponent , is zero or a subdirect product of , and is zero or a subdirect product of .
49 citations
TL;DR: In this paper, the DMP inverse is extended from matrices to operators and many closely equivalent relations among these inverses are investigated by using appropriate idempotents, and some new properties of the inverse are obtained and some known results are generalized.
Abstract: Let $$\mathcal {H}$$H be a Hilbert space The recently introduced notions of the DMP inverse are extended from matrices to operators The group, Moore---Penrose, Drazin inverses are integrated by DMP inverse and many closely equivalent relations among these inverses are investigated by using appropriate idempotents Some new properties of DMP inverse are obtained and some known results are generalized
31 citations
TL;DR: In this paper, the connection between Gabor frames for quasicrystals, the topology of the hull of a quasICrystal Λ, and the K-theory of an associated twisted groupoid algebra was studied.
22 citations
TL;DR: It is shown that idempotent uninorms are in one-to-one correspondence with special linear orders on 0 , 1 and the characterizing multi-function of the uninorm is investigated.
19 citations
TL;DR: In this paper, the rank and idempotent rank of a semigroup E(X,P) generated by the idempots of the semigroup T (X and P) were derived.
Abstract: We calculate the rank and idempotent rank of the semigroup E(X,P) generated by the idempotents of the semigroup T (X,P), which consists of all transformations of the finite set X preserving a non-uniform partition P. We also classify and enumerate the idempotent generating sets of this minimal possible size. This extends results of the first two authors in the uniform case.
19 citations
TL;DR: In this article, the Brauer algebra Brd(δ) over the complex numbers for an integral parameter δ can be equipped with a grading, which is moreover Morita equivalent to a Koszul algebra.
Abstract: We show that the Brauer algebra Brd(δ) over the complex numbers for an integral parameter δ can be equipped with a grading. In case δ ≠ 0 it becomes a graded quasi-hereditary algebra which is moreover Morita equivalent to a Koszul algebra. These results are obtained by realizing the Brauer algebra as an idempotent truncation of a certain level two VW-algebra ⩔ cycl d (N) for some large positive integral parameter N . The parameter δ appears here in the choice of a cyclotomic quotient. This cyclotomic VW-algebra arises naturally as an endomorphism algebra of a certain projective module in parabolic category O of type D. In particular, the graded decomposition numbers are given by the associated parabolic Kazhdan-Lusztig polynomials.
17 citations
TL;DR: In this article, a constrained evolution algebra of bisexual population was introduced by identifying the coefficients of inheritance of a bisexual population as the structure constants of the algebra, and the authors studied dynamical systems generated by the quadratic evolution operators of corresponding constrained algebras.
17 citations
TL;DR: In this paper, the quantum version of the integration over the diagonal subgroup is analyzed, and it is shown that the corresponding integration represented by a certain idempotent state on C(𝔾) makes sense as long as C is of Kac type.
Abstract: By analogy with the classical construction due to Forrest, Samei and Spronk, we associate to every compact quantum group 𝔾, a completely contractive Banach algebra AΔ(𝔾), which can be viewed as a deformed Fourier algebra of 𝔾. To motivate the construction, we first analyze in detail the quantum version of the integration over the diagonal subgroup, showing that although the quantum diagonal subgroups in fact never exist, as noted earlier by Kasprzak and Soltan, the corresponding integration represented by a certain idempotent state on C(𝔾) makes sense as long as 𝔾 is of Kac type. Finally, we analyze as an explicit example the algebras AΔ(ON+), N ≥ 2, associated to Wang’s free orthogonal groups, and show that they are not operator weakly amenable.
16 citations
TL;DR: In this paper, the elements of the (maximum) idempotent generated subsemigroup of the Kauffman monoid were characterized in terms of combinatorial data associated to certain normal forms.
Abstract: We characterise the elements of the (maximum) idempotent generated subsemigroup of the Kauffman monoid in terms of combinatorial data associated to certain normal forms. We also calculate the smallest size of a generating set and idempotent generating set.
TL;DR: This short note finds that Δ f is computed incorrectly in their paper (leading to Δ h ? 2 21 + 1 2 ) and correct their result by disproving the related conjecture for the 21-variable balanced functions.
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TL;DR: The study of the local structure of idempotent finite algebras is refined, demonstrating that the edges of the graph of an algebra can be made `thin', that is, there are term operations that behave very similar to semilattice, majority, or affine operations on 2-element subsets of the algebra.
Abstract: We refine and advance the study of the local structure of idempotent finite algebras started in [A.Bulatov, The Graph of a Relational Structure and Constraint Satisfaction Problems, LICS, 2004]. We introduce a graph-like structure on an arbitrary finite idempotent algebra omitting type 1. We show that this graph is connected, its edges can be classified into 3 types corresponding to the local behavior (semilattice, majority, or affine) of certain term operations, and that the structure of the algebra can be `improved' without introducing type 1 by choosing an appropriate reduct of the original algebra. Then we refine this structure demonstrating that the edges of the graph of an algebra can be made `thin', that is, there are term operations that behave very similar to semilattice, majority, or affine operations on 2-element subsets of the algebra. Finally, we prove certain connectivity properties of the refined structures.
This research is motivated by the study of the Constraint Satisfaction Problem, although the problem itself does not really show up in this paper.
TL;DR: This paper explores min-plus methods for continuous-time stochastic control on a finite-time horizon and obtains an algorithm for recursive computation of the time-discretized values under the idempotent distributed dynamic programming principle (IDDPP).
Abstract: Max-plus methods have previously been used to solve deterministic control problems. The methods are based on max-plus (or min-plus) expansions and can yield curse-of-dimensionality-free numerical methods. In this paper, we explore min-plus methods for continuous-time stochastic control on a finite-time horizon. We first approximate the original value function via time-discretization. By generalizing the min-plus distributive property to continuum spaces, we obtain an algorithm for recursive computation of the time-discretized values, which we refer to as the idempotent distributed dynamic programming principle (IDDPP). Under the IDDPP, the value function at each step can be represented as an infimum of functions in a certain class. This is a min-plus expansion for the value function. For the specific class of problems considered here, we see that the class can be taken as that consisting of the quadratic functions. A means for reducing the numbers of constituent quadratic functions is discussed.
TL;DR: In this article, a clustering algebra structure of multihomogeneous coordinate rings of partial flag varieties of arbitrary Dynkin type using Cohen-Macaulay modules over orders is described.
Abstract: We describe a categorification of the cluster algebra structure of multihomogeneous coordinate rings of partial flag varieties of arbitrary Dynkin type using Cohen–Macaulay modules over orders. This completes the categorification of Geiss, Leclerc and Schroer by adding the missing coefficients. To achieve this, for an order A and an idempotent e∈A, we introduce a subcategory CMeA of CMA and study its properties. In particular, under some mild assumptions, we construct an equivalence of exact categories (CMeA)∕[Ae]≅SubQ for an injective B-module Q, where B:=A∕(e). These results generalize work by Jensen, King and Su concerning the cluster algebra structure of the Grassmannian Grm(ℂn).
TL;DR: In this paper, a well-behaved transfer map from the p-local Burnside ring of the underlying p-group S to the local Burnside rings of a saturated fusion system F was constructed, which is the unique idempotent of S satisfying properties of Linckelmann and Webb.
TL;DR: In this article, the authors introduced Rees semigroups of directed graphs, a generalization of Rees matrix semiigroups and incidence semigroup of digraphs, and proved that all ideals in the semigroup semiring of a digraph have the largest possible weight.
Abstract: Recent research has motivated the investigation of the weights of ideals in semiring constructions based on semigroups. The present paper introduces Rees semigroups of directed graphs. This new construction is a common generalization of Rees matrix semigroups and incidence semigroups of digraphs. For each finite subsemigroup $$S$$
of the Rees semigroup of a digraph and for every zero-divisor-free idempotent semiring $$F$$
with identity element, our main theorem describes all ideals $$J$$
in the semigroup semiring $$F_0[S]$$
such that $$J$$
has the largest possible weight.
TL;DR: In this paper, it was shown that a nonzero idempotent is zero-diagonal if and only if it is not a Hilbert-Schmidt perturbation of a projection, along with other useful equivalences.
Abstract: We prove that a nonzero idempotent is zero-diagonal if and only if it is not a Hilbert-Schmidt perturbation of a projection, along with other useful equivalences. Zero-diagonal operators are those whose diagonal entries are identically zero in some basis.
We also prove that any bounded sequence appears as the diagonal of some idempotent operator, thereby providing a characterization of inner products of dual frame pairs in infinite dimensions. Furthermore, we show that any absolutely summable sequence whose sum is a positive integer appears as the diagonal of a finite rank idempotent.
26 Jul 2016
TL;DR: In this paper, the relative rank of all transformations on a finite set X with restricted range Y is derived for all order-preserving transformations in T(X, Y ) modulo the semigroup of all extensions of the bijections on Y, modulo all idempotent transformations in X; Y, and modulo T(T, X, Y ).
Abstract: In this paper, we determine the relative rank of the semigroup T(X; Y ) of all transformations on a nite set X with restricted range Y modulo the semigroup of all extensions of the bijections on Y , modulo the idempotent order-preserving transformations in T(X; Y ), and modulo the semigroup of all order-preserving transformations in T(X; Y ).
TL;DR: For an arbitrary left Artinian ring R, explicit descriptions of all the left denominator sets S of R and left localizations S−1R of R are given in this article.
Abstract: For an arbitrary left Artinian ring R, explicit descriptions are given of all the left denominator sets S of R and left localizations S−1R of R. It is proved that, up to R-isomorphism, there are only finitely many left localizations and each of them is an idempotent localization, i.e. S−1R ≃ S e−1R and ass(S) = ass(Se) where Se = {1,e} is a left denominator set of R and e is an idempotent. Moreover, the idempotent e is unique up to a conjugation. It is proved that the number of maximal left denominator sets of R is finite and does not exceed the number of isomorphism classes of simple left R-modules. The set of maximal left denominator sets of R and the left localization radical of R are described.
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TL;DR: In this article, the fundamental relation on a regular multiplicative hyperring R = R/? of R is proved to be a regular ring, and the regular hyperideal of R =R/? is shown to be regular.
Abstract: We introduce and study regular multiplicative hyperrings, as a generalization of classical rings Also, we use the fundamental relation ? on a given regular multiplicative hyperring R and prove that the fundamental ring R=R/? of R is a regular ring Finally, we investigate the algebraic properties of M(R), the regular hyperideal of R, generated by all elements of R such that its generated hyperideals is regular
TL;DR: In this article, the authors present a way of computing asymptotic expansions of solutions to algebraic and dif-ferential equations and present a survey of some of its applications.
Abstract: Here we present a way of computation of asymptotic expansions of solutions to algebraic and dif- ferential equations and present a survey of some of its applications. The way is based on ideas and algorithms of Power Geometry. Power Geometry has applications in Alge- braic Geometry, Differential Algebra, Nonstandard Analysis, Microlocal Analysis, Group Analysis, Tropical/Idempotent Mathematics and so on. We also discuss a connection of Power Geometry with Idempotent Mathematics.
TL;DR: Under certain countable setting, the main result roughly says that every nondegenerate Ramsey algebra has a nonprincipal idempotent ultrafilter in some nontrivial countable field of sets.
Abstract: Hindman's Theorem says that every finite coloring of the positive natural numbers has a monochromatic set of finite sums. Ramsey algebras, recently introduced, are structures that satisfy an analogue of Hindman's Theorem. It is an open problem posed by Carlson whether every Ramsey algebra has an idempotent ultrafilter. This paper develops a general framework to study idempotent ultrafilters. Under certain countable setting, the main result roughly says that every nondegenerate Ramsey algebra has a nonprincipal idempotent ultrafilter in some nontrivial countable field of sets. This amounts to a positive result that addresses Carlson's question in some way.
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TL;DR: In this paper, it was shown that an idempotent variety has a cube term if and only if its free algebra on two generators has no $d$-ary compatible cross.
Abstract: We show that an idempotent variety has a $d$-dimensional cube term if and only if its free algebra on two generators has no $d$-ary compatible cross. We employ Hall's Marriage Theorem to show that a variety of finite signature whose fundamental operations have arities $n_1, \ldots, n_k$ has a $d$-dimensional cube term if and only if it has one of dimension $d=1+\sum_{i=1}^k (n_i-1)$. This lower bound on dimension is shown to be sharp. We show that a pure cyclic term variety has a cube term if and only if it contains no $2$-element semilattice. We prove that the Maltsev condition "existence of a cube term" is join prime in the lattice of idempotent Maltsev conditions.
TL;DR: In this article, it was shown that if the difference of two measurable idempotents is a positive operator, then this difference is a projection, and that a semihyponormal measurable tripotent is the difference between two orthogonal projections.
Abstract: New properties of the space of integrable (with respect to the faithful normal semifinite trace) operators affiliated with a semifinite von Neumann algebra are found. A trace inequality for a pair of projections in the von Neumann algebra is obtained, which characterizes trace in the class of all positive normal functionals on this algebra. A new property of a measurable idempotent are determined. A useful factorization of such an operator is obtained; it is used to prove the nonnegativity of the trace of an integrable idempotent. It is shown that if the difference of two measurable idempotents is a positive operator, then this difference is a projection. It is proved that a semihyponormal measurable idempotent is a projection. It is also shown that a hyponormal measurable tripotent is the difference of two orthogonal projections.
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TL;DR: In this article, a 1-1 correspondence between idempotent states on a locally compact quantum group G and integrable coideals in the von Neumann algebra of bounded measurable functions on G that are preserved by the scaling group is established.
Abstract: We establish a one to one correspondence between idempotent states on a locally compact quantum group G and integrable coideals in the von Neumann algebra of bounded measurable functions on G that are preserved by the scaling group. In particular we show that there is a 1-1 correspondence between idempotent states on G and psi-expected left invariant von Neumann subalgebras of bounded measurable functions on G. We characterize idempotent states of Haar type as those corresponding to integrable normal coideals preserved by the scaling group. We also establish a one to one correspondence between open subgroups of G and convolutionally central idempotent states on the dual of G. Finally we characterize coideals corresponding to open quantum subgroups of G as those that are normal and admit an atom. As a byproduct of this study we get a number of universal lifting results for Podles condition, normality and regularity and we generalize number of results known before to hold under the coamenability assumption.
TL;DR: In this paper, it was shown that if two idempotent rings R and S are Morita equivalent, then for every von Neumann regular element a ∈ R the local algebra of R at a, R a, is isomorphic to the one of S for some natural n and some idemomorphism u in
Abstract: In this paper we prove that if two idempotent rings R and S are Morita equivalent then for every von Neumann regular element a ∈ R the local algebra of R at a, R a , is isomorphic to \(\mathbb{M}_{n}(S)_{u}\) for some natural n and some idempotent u in \(\mathbb{M}_{n}(S)\). We give examples showing that the converse of this result is not true in general and establish the converse for σ-unital rings having a σ-unit consisting of von Neumann regular elements.Our next aim is to prove that, for idempotent rings, a property is Morita invariant if it is invariant under taking local algebras at von Neumann regular elements and under taking matrices.The previous results are used to check the Morita invariance of certain ring properties (being locally left/right artinian/noetherian, being categorically left/right artinian, being an I0-ring and being properly purely infinite) and of certain graph properties in the context of Leavitt path algebras (Condition (L), Condition (K) and cofinality). A different proof of the fact that a graph with an uncountable emitter does not admit a desingularization is also given.
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TL;DR: In this paper, the sub-power membership problem (SMP) for finite semigroups was shown to be NP-complete for bands, and the greatest variety of bands all of whose finite members induce a tractable SMP was determined.
Abstract: Fix a finite semigroup $S$ and let $a_1,\ldots,a_k, b$ be tuples in a direct power $S^n$. The subpower membership problem (SMP) for $S$ asks whether $b$ can be generated by $a_1,\ldots,a_k$. For bands (idempotent semigroups), we provide a dichotomy result: if a band $S$ belongs to a certain quasivariety, then $SMP(S)$ is in P; otherwise it is NP-complete.
Furthermore we determine the greatest variety of bands all of whose finite members induce a tractable SMP. Finally we present the first example of two finite algebras that generate the same variety and have tractable and NP-complete SMPs, respectively.
TL;DR: In this paper, the abstract concavity and support set of extended valued elementary topical functions are characterized for the upward and downward sets of a b-complete idempotent semimodule.
Abstract: In this article, we study topical functions f:X→K defined on a b-complete idempotent semimodule X over a b-complete idempotent semifield K with values in K. We characterize the abstract concavity and support set of this class of functions. Next, we investigate the abstract concavity of extended valued topical functions , where and ⊤: = supK. Finally, as an application, we present characterizations of upward and downward sets by using extended valued elementary topical functions.
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TL;DR: In this article, the notion of separability idempotent in the C*-algebra framework was studied, analogous to the notion in the purely algebraic setting, typically considered in the case of (finite-dimensional) algebras with identity, then later also considered in multiplier algebra framework by the second-named author.
Abstract: In this paper, we study the notion of a separability idempotent in the C*-algebra framework. This is analogous to the notion in the purely algebraic setting, typically considered in the case of (finite-dimensional) algebras with identity, then later also considered in the multiplier algebra framework by the second-named author. The current work was motivated by the appearance of such objects in the authors' ongoing work on locally compact quantum groupoids.