Showing papers on "Idempotence published in 2011"
TL;DR: A collection of all soft sets with respect to new operations give rise to four idempotent monoids, which with the help of these monoids can study semiring (hemiring) structures of soft sets.
Abstract: Recently new operations have been defined for soft sets. In this paper, we study some important properties associated with these new operations. A collection of all soft sets with respect to new operations give rise to four idempotent monoids. Then with the help of these monoids we can study semiring (hemiring) structures of soft sets. Some of these semirings (hemirings) are actually lattices. Finally, we show that soft sets with a fixed set of parameters are MV algebras and BCK algebras.
151 citations
TL;DR: The main goal of as discussed by the authors is to prove that the idempotent completions of triangulated categories of singularities of two schemes are equivalent if the formal completions along singularities are isomorphic.
103 citations
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TL;DR: This article showed that the walled Brauer algebra is Morita equivalent to an idempotent truncation of a certain infinite dimensional version of Khovanov's arc algebra, as suggested by recent work of Cox and De Visscher.
Abstract: We introduce some graded versions of the walled Brauer algebra, working over a field of characteristic zero. This allows us to prove that the walled Brauer algebra is Morita equivalent to an idempotent truncation of a certain infinite dimensional version of Khovanov's arc algebra, as suggested by recent work of Cox and De Visscher. We deduce that the walled Brauer algebra is Koszul whenever its defining parameter is non-zero.
68 citations
TL;DR: The singular part of the partition monoid is studied, that is, the ideal , where is the symmetric group and its rank and idempotent-rank are both equal to .
Abstract: We study the singular part of the partition monoid ; that is, the ideal , where is the symmetric group. Our main results are presentations in terms of generators and relations. We also show that is idempotent generated, and that its rank and idempotent-rank are both equal to . One of our presentations uses an idempotent generating set of this minimal cardinality.
64 citations
TL;DR: There are proven generalizations of the Holder’s and Minkowski's inequalities for the pseudo-integral for the real semiring with pseudo-operations.
59 citations
TL;DR: In this article, the authors introduce the notion of chain of evolution algebras, which is a class of non-isomorphic evolution algebraic structures that have a unique (resp. infinitely many) absolute nilpotent element at a fixed time.
59 citations
TL;DR: In this paper, the star partial order on B (H) was defined using self-adjoint idempotent operators, and some properties of the generalized concept of order relations on B(H) were investigated.
40 citations
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TL;DR: In this article, it was shown that all rings derived equivalent to Q-factorial terminalizations in dimension three are MMAs, and that any linear algebra A is equivalent to X if and only if A is an MMAs.
Abstract: Let X be a Gorenstein normal 3-fold satisfying (ELF) with local rings which are at worst isolated hypersurface (e.g. terminal) singularities. By using the singular derived category D_{sg}(X) and its idempotent completion, we give necessary and sufficient categorical conditions for X to be Q-factorial and complete locally Q-factorial respectively. We then relate this information to maximal modification algebras(=MMAs), introduced in [IW10], by showing that if an algebra A is derived equivalent to X as above, then X is Q-factorial if and only if A is an MMA. Thus all rings derived equivalent to Q-factorial terminalizations in dimension three are MMAs. As an application, we extend some of the algebraic results in Burban-Iyama-Keller-Reiten [BIKR] and Dao-Huneke [DH] using geometric arguments.
30 citations
TL;DR: In this article, the pseudo-amenability of semigroup algebra l1(S) is investigated, where S is an inverse semigroup with uniformly locally finite idempotent set.
Abstract: In this paper, we investigate the pseudo-amenability of semigroup algebra l1(S), where S is an inverse semigroup with uniformly locally finite idempotent set. In particular, we show that for a Brandt semigroup \(S={\mathcal{M}}^{0}(G,I)\), the pseudo-amenability of l1(S) is equivalent to the amenability of G.
25 citations
DOI•
06 Jul 2011
TL;DR: In this article, it was shown that a 2-copula is invariant under if for almost all, being the characteristic function of the transition probability function of a set, and that the sets invariant to almost all transformations form a sub-algebra of the Lebesgue measurable sets.
Abstract: A 2-copula induces a transition probability function via where , denoting the Lebesgue measurable subsets of . We say that a set is invariant under if for almost all , being the characteristic function of . The sets invariant under form a sub- -algebra of theLebesgue measurable sets, which we denote . A set is called an atom if it has positive measure and if for any , is either or 0. A 2-copula is idempotent if . Here denotes the product defined in [1]. Idempotent 2-copulas are classified and characterized asfollows: (i) An idempotent is said to be nonatomic if contains noatoms. If is a nonatomic idempotent, then it is the product of a leftinvertible copula and its transpose. That is, there exists a copula such that where (ii) An idempotent is said to be totally atomic if there exist essentiallydisjoint atoms with If is a totally atomic idempotent, then it is conjugate to an ordinal sumof copies of the product copula. That is, there exists a copula satisfying and a partition of such that \begin{equation}F=C*(\oplus _{\cal P}F_k)*C^T \end{equation} where eachcomponent in the ordinal sum is the product copula . (iii) An idempotent is said to be atomic (but not totally atomic) if contains atoms but the sum of the measures of a maximal collection ofessentially disjoint atoms is strictly less than 1. In this mixed case, thereexists a copula invertible with respect to and a partition of for which (1) holds, with being a nonatomic idempotent copula andwith for . Some of the immediate consequences of this characterization are discussed.
23 citations
TL;DR: In this paper, the group inverse of aP + bQ + cPQ + dQP + ePQP+ ePQLP + fQPQ+ gPQPLPQ
Abstract: In this article, we discuss the group inverse of aP + bQ + cPQ + dQP + ePQP + fQPQ + gPQPQ of idempotent matrices P and Q, where a, b, c, d, e, f, g ∈ ℂ and a ≠ 0, b ≠ 0, put forward its explicit expressions, and some necessary and sufficient conditions for the existence of the group inverse of aP + bQ + cPQ.
TL;DR: In this paper, the extremal property of the eigenvalue of an irreducible matrix in idempotent algebra is studied and it is shown that this value is the minimum value of some functional defined using this matrix on the set of vectors with nonzero components.
Abstract: An extremal property of the eigenvalue of an irreducible matrix in idempotent algebra is studied. It is shown that this value is the minimum value of some functional defined using this matrix on the set of vectors with nonzero components. The minimax problem of location of a single facility (the Rawls problem) on a plane with rectilinear distance is considered. For this problem, we give the corresponding representation in terms of idempotent algebra and suggest a new algebraic solution, which is based on the results of investigation of the extremal property of eigenvalue and reduces to finding the eigenvalue and eigenvectors of a certain matrix.
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TL;DR: For | A |=2, the aim of this paper is to provide a better understanding of this uncountable lattice of equational classes of Boolean functions, by analyzing its “closed" intervals, for idempotent classes C 1 and C 2.
Abstract: Let A be a finite set with at least two elements. The composition of two classes I and J of operations on A, is defined as the set of all compositions of functions in I with functions in J. This binary operation gives a monoid structure to the set E_A of all equational classes of operations on A. The set E_A of equational classes of operations on A also constitutes a complete distributive lattice under intersection and union. Clones of operations, i.e. classes containing all projections and idempotent under class composition, also form a lattice which is strictly contained in E_A. In the Boolean case |A|=2, the lattice E_A contains uncountably many equational classes, but only countably many of them are clones. The aim of this paper is to provide a better understanding of this uncountable lattice of equational classes of Boolean functions, by analyzing its "closed" intervals" [C_1,C_2], for clones C_1 and C_2. For |A|=2, we give a complete classification of all such closed intervals in terms of their size, and provide a simple, necessary and sufficient condition characterizing the uncountable closed intervals of E_A.
TL;DR: A method to construct piecewise linear binary aggregation functions on the unit interval, based on a triangulation of the unit square with one additional vertex is introduced, which can also be used to approximate binary aggregation Functions.
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TL;DR: In this article, a complete description of subdirectly irreducible state BL-algebras as well as of sub-directly irreproducible state-morphism pseudo-MTL-algeses is given.
Abstract: We present a complete description of subdirectly irreducible state BL-algebras as well as of subdirectly irreducible state-morphism BL-algebras. In addition, we present a general theory of state-morphism algebras, that is, algebras of general type with state-morphism which is an idempotent endomorphism. We define a diagonal state-morphism algebra and we show that every subdirectly irreducible state-morphism algebra can be embedded into a diagonal one. We describe generators of varieties of state-morphism algebras, in particular ones of state-morphism BL-algebras, state-morphism MTL-algebras, state-morphism non-associative BL-algebras, and state-morphism pseudo MV-algebras.
Journal Article•
TL;DR: In this paper, the extremal property of the eigenvalue of an irreducible matrix in idempotent algebra is studied and it is shown that this value is the minimum value of some functional defined using this matrix on the set of vectors with nonzero components.
Abstract: An extremal property of the eigenvalue of an irreducible matrix in idempotent algebra is studied. It is shown that this value is the minimum value of some functional defined using this matrix on the set of vectors with nonzero components. The minimax problem of location of a single facility (the Rawls problem) on a plane with rectilinear distance is considered. For this problem, we give the corresponding representation in terms of idempotent algebra and suggest a new algebraic solution, which is based on the results of investigation of the extremal property of eigenvalue and reduces to finding the eigenvalue and eigenvectors of a certain matrix.
Journal Article•
TL;DR: A new algebraic approach based on investigation of extremal properties of eigenvalues for irreducible matrices is developed to solve multidimensional problems that involve minimization of functionals defined on idempotent vector semimodules.
Abstract: Minimax single facility location problems in multidimensional space with Chebyshev distance are examined within the framework of idempotent algebra. The aim of the study is twofold: first, to give a new algebraic solution to the location problems, and second, to extend the area of application of idempotent algebra. A new algebraic approach based on investigation of extremal properties of eigenvalues for irreducible matrices is developed to solve multidimensional problems that involve minimization of functionals defined on idempotent vector semimodules. Furthermore, an unconstrained location problem is considered and then represented in the idempotent algebra settings. A new algebraic solution is given that reduces the problem to evaluation of the eigenvalue and eigenvectors of an appropriate matrix. Finally, the solution is extended to solve a constrained location problem.
TL;DR: A very brief introduction to mathematics of semirings (including idempotent and tropical mathematics) is presented and concrete applications to optimization problems, idem Potent linear algebra and interval analysis are indicated.
Abstract: This isaut]Grigory L. Litvinovaut]Victor P. Maslovaut]Anatoly Ya. Rodionovaut]Andrei N. Sobolevskii a survey paper on applications of mathematics of semirings to numerical analysis and computing. Concepts of universal algorithm and generic program are discussed. Relations between these concepts and mathematics of semirings are examined. A very brief introduction to mathematics of semirings (including idempotent and tropical mathematics) is presented. Concrete applications to optimization problems, idempotent linear algebra and interval analysis are indicated. It is known that some nonlinear problems (and especially optimization problems) become linear over appropriate semirings with idempotent addition (the so-called idempotent superposition principle). This linearity over semirings is convenient for parallel computations.
TL;DR: This first paper studies in detail block splitting operators and their lattice-theoretical and monoid properties; in particular it considers their idempotence (a requirement in image segmentation) and describes block splitting openings as operators splitting each block into its connected components according to a partial connection.
Abstract: Image segmentation algorithms can be modelled as image-guided operators (maps) on the complete lattice of partitions of space, or on the one of partial partitions (i.e., partitions of subsets of the space). In particular region-splitting segmentation algorithms correspond to block splitting operators on the lattice of partial partitions, in other words anti-extensive operators that act by splitting each block independently. This first paper studies in detail block splitting operators and their lattice-theoretical and monoid properties; in particular we consider their idempotence (a requirement in image segmentation). We characterize block splitting openings (kernel operators) as operators splitting each block into its connected components according to a partial connection; furthermore, block splitting openings constitute a complete sublattice of the complete lattice of all openings on partial partitions. Our results underlie the connective approach to image segmentation introduced by Serra. The second paper will study two classes of non-isotone idempotent block splitting operators, that are also relevant to image segmentation.
TL;DR: In this paper, a lattice-ordered ring of real functions of a frame L and its subclasses LSC(L) and USC(LSC) of lower and upper semicontin- uous real functions is studied.
TL;DR: Here, it is seen that idempotent methods may be used to solve some classes of stochastic control problems, and the key is the use of the idempotsent distributive property.
TL;DR: A classification of all four-dimensional power-commutative real division algebras is given in this paper, which leads to a complete and irredundant classification.
Abstract: A classification of all four-dimensional power-commutative real division algebras is given. It is shown that every four-dimensional power-commutative real division algebra is an isotope of a particular kind of a quadratic division algebra. The description of such isotopes in dimensions four and eight is reduced to the description of quadratic division algebras. In dimension four, this leads to a complete and irredundant classification. As a special case, the finite-dimensional power-commutative real division algebras that have a unique non-zero idempotent are characterized.
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TL;DR: For weakly idempotent complete additive categories, the authors showed that the stable exact sequences define a maximal exact structure (in the sense of Quillen) in any pre-abelian category.
Abstract: Sieg and Wegner showed that the stable exact sequences define a maximal exact structure (in the sense of Quillen) in any pre-abelian category. We generalize this result for weakly idempotent complete additive categories.
TL;DR: An extension of Ulam's problem, known in the literature as the [email protected]?vruta-Rassias stability problem, is answered affirmatively for this equation and an approximate property of continuous Banach space projections is obtained.
TL;DR: In this article, the Max-Plus convexity separation can be achieved by a single map if one considers linear maps with values in a linearly ordered semimodule, whose construction is given here.
TL;DR: In this article, Johnson et al. characterized the norm one idempotents in M cb A(G): the indicator function of a set E ⊂ G is defined to be a norm one ǫ-idempotent if and only ifE is a coset of an open subgroup of G. As applications, they described the closed idealsof A Mcb (G) with an approximate identity bounded by 1.
Abstract: For a locally compact group G, let A(G) be its Fourieralgebra, let M cb A(G) denotethe completely bounded multipliers of A(G), and let A Mcb (G) stand for the closureof A(G) in M cb A(G). We characterize the norm one idempotents in M cb A(G): theindicator function of a set E ⊂ G is a norm one idempotent in M cb A(G) if and only ifE is a coset of an open subgroup of G. As applications, we describe the closed idealsof A Mcb (G) with an approximate identity bounded by 1, and we characterize those Gfor which A Mcb (G) is 1-amenable in the sense of B. E. Johnson. (We can even slightlyrelax the norm bounds.) Keywords : amenability; bounded approximate identity; cb-multiplier norm; Fourier algebra; normone idempotent.2000 Mathematics Subject Classification : Primary 43A22; Secondary 20E05; 43A30, 46J10, 46J40,46L07, 47L25. Introduction The Fourier algebra A(G) and Fourier–Stieltjes algebra B(G) of a locally compact groupG were introduced by P. Eymard in [Eym]. If G is abelian with dual group Gˆ, thesealgebras are isometrically isomorphic to L
TL;DR: In this article, it was shown that for smooth manifolds X and Y, any isomorphism between the algebras of generalized functions (in the sense of Colombeau) on X is given by composition with a unique generalized function from Y to X.
Abstract: We show that for smooth manifolds X and Y, any isomorphism between the algebras of generalized functions (in the sense of Colombeau) on X, resp. Y is given by composition with a unique generalized function from Y to X. We also characterize the multiplicative linear functionals from the Colombeau algebra on X to the ring of generalized numbers. Up to multiplication with an idempotent generalized number, they are given by an evaluation map at a compactly supported generalized point on X.
TL;DR: In this paper, the universal similarity factorization equalities for linear combinations of two commutative involutory and two idempotent matrices and their products were derived for a variety of applications.
Abstract: A square matrix A of order n is said to be involutory if A 2 = I n , and to be idempotent if A 2 = A. In this article, we give two universal similarity factorization equalities for linear combinations of two commutative involutory and two idempotent matrices and their products. As applications, we derive some disjoint decompositions for these linear combinations, and use the disjoint decompositions to derive a variety of results on the determinants, ranks, traces, inverses, generalized inverses and similarity decompositions of these linear combinations. In particular, we present some collections of involutory, idempotent and tripotent matrices generated from these linear combinations.
TL;DR: In this paper, the authors studied the class of algebras satisfying an identity of the shape aX2X2+bX4−λX3−μX2−νX, and showed that these classes admit principal and plenary train identities of rank ≤ 4.
Abstract: We continue the study of the algebras satisfying an identity of the shape aX2X2 + bX4 − λX3 − μX2 − νX, we introduce here the case a + b = 1, ab ≠ 0, λ = 2, and μ + ν = −1. By studying a 26 × 27 linear system, we show that this class admits a partition into six parts among which four correspond to algebras not verifying other identity of degree ≤4. In this article we study these four algebras. We show they satisfy principal and plenary train identities of rank > 4, we determine the algebras admitting an idempotent and we give their Peirce decompositions in the two cases: with and without idempotent.
TL;DR: In this article, the authors present necessary and sufficient conditions under which an arbitrary dimonoid is a semilattice of archimedean subdimonoids, and show that an idempotent decomposition of such a decomposition can be obtained.
Abstract: We present some congruence on the dimonoid with an idempotent operation and use it to obtain semilattice decompositions of an idempotent dimonoid. Also we give necessary and sufficient conditions under which an arbitrary dimonoid is a semilattice of archimedean subdimonoids.