Showing papers on "Idempotence published in 2005"
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TL;DR: In this paper, a brief introduction to idempotent and tropical mathematics can be found, which can be treated as a result of the so-called Maslov dequantization of the traditional mathematics over numerical fields as the Planck constant tends to zero taking imaginary values.
Abstract: This paper is a brief introduction to idempotent and tropical mathematics. Tropical mathematics can be treated as a result of the so-called Maslov dequantization of the traditional mathematics over numerical fields as the Planck constant $\hbar$ tends to zero taking imaginary values.
54 citations
TL;DR: Kleene’s result for formal tree series over a commutative semiring A is proved, i.e., the class of formal treeseries over A which are accepted by weighted tree automata, and theclass of rational tree seriesover A are equal.
Abstract: In this paper we prove Kleene’s result for formal tree series over a commutative semiring A (which is not necessarily complete or continuous or idempotent), i.e., the class of formal tree series over A which are accepted by weighted tree automata, and the class of rational tree series over A are equal. We show the result by direct automata-theoretic constructions and prove their correctness.
47 citations
18 Oct 2005
TL;DR: In this article, it was shown that the invertibility of the linear combination λ 1 P + λ 2 Q is independent of the choice of λ i, i = 1, 2.
Abstract: Let P and Q be two idempotents on a Hilbert space. In this note, we prove that the invertibility of the linear combination λ 1 P + λ 2 Q is independent of the choice of λ i , i = 1, 2, if λ 1 λ 2 ≠ 0 and λ 1 + λ 2 ≠ 0.
32 citations
TL;DR: In this paper, the multiplicative reduct of an idempotent semiring with commutative addition is a regular band and the lattice generated by these 13 semiring varieties is described and models for the semirings free in these varieties are given.
Abstract: The multiplicative reduct of an idempotent semiring with commutative addition is a regular band. Accordingly there are 13 distinct varieties consisting of idempotent semirings with commutative addition corresponding to the 13 subvarieties of the variety of regular bands. The lattice generated by the these 13 semiring varieties is described and models for the semirings free in these varieties are given.
26 citations
TL;DR: It is shown that Greibach’s normal form theorem depends only on a few equational properties of least pre-fixed points in semirings, and eliminations of chain and deletion rules depend on their inequational properties (and the idempotence of addition).
24 citations
TL;DR: An elementary proof of a theorem first proved by J. A. Erdos, which states that every noninvertible n × n matrix is a finite product of matrices M with the property that M = M .
Abstract: In this note we give an elementary proof of a theorem first proved by J. A. Erdos [3]. This theorem, which is the main result of [3], states that every noninvertible n × n matrix is a finite product of matrices M with the property that M = M . (These are known as idempotent matrices. Noninvertible matrices are also called singular matrices.) An alternative formulation of this result reads: every noninvertible linear mapping of a finite dimensional vector space is a finite product of idempotent linear mappings α, linear mappings that satisfy α = α. This result was motivated by a result of J. M. Howie asserting that each selfmapping α of a nonempty finite set X with image size at most |X| − 1 (which occurs precisely when α is noninvertible) is a product of idempotent mappings. We shall see that Erdos’s theorem is a consequence of Howie’s result. Together the papers [3] and [4] are cited in over one hundred articles, dealing with subjects including universal algebra, ring theory, topology, and combinatorics. Since its publication, various proofs of the result in [3] have appeared. For example, a semigroup theoretic proof appears in [1, pp. 121-131] and linear operator theory is used to prove the theorem in [2]. Here we give a new proof using a basic combinatorial argument. Unlike the previous proofs our argument involves only elementary results from linear algebra and one basic result concerning permutations. On the way to proving the main result of this note we provide a short proof of Howie’s result. Throughout this paper X signifies an arbitrary nonempty finite set. If α : A → X, where A is a subset of X, then A is the domain of α; we denote this set by dom(α). Naturally, the set α(A) is called the image of α and is denoted by im(α). Recall that a mapping α is injective (or one-to-one) if α(x) 6= α(y) for all x and y in dom(α) with x 6= y. Let TX denote the set of all mappings from X to X with domain X. We note that this set is closed under composition of mappings and that this composition is associative. We now define one of the most important notions we require in the proofs in this note. For a mapping α : dom(α) → X we say that α is a restriction of an element β of TX if β and α agree on the domain of α. In other words, β(x) = α(x) for all x in dom(α). For x and y in X we denote the transposition that fixes every point of X other than x or y and that maps x to y, and vice versa, by (x y).
22 citations
TL;DR: A primitive symmetric association scheme of class 2 is naturally embedded as a two-distance set in the unit sphere of Euclidean space, with respect to the primitive idempotent E1 of the Bose-Mesner algebra of the association scheme.
Abstract: A primitive symmetric association scheme of class 2 is naturally embedded as a two-distance set in the unit sphere of Euclidean space, with respect to the primitive idempotent E1 of the Bose-Mesner algebra of the association scheme. Then it is shown that the ratio of the two distances of the two-distance set is instantly read from the character table (i.e., the first eigen matrix P) of the association scheme.
20 citations
TL;DR: The problem of characterizing linear operators on matrix algebras that leave invariant certain functions, subsets or relations has attracted the attention of many mathematicians (see survey papers as mentioned in this paper for details).
19 citations
01 Jan 2005
TL;DR: In this paper, an approach based on approximation of the matrix of the system by means of matrices of simple structure is applied to evaluate bounds on the mean rate of growth of the state vector of a dynamical system.
Abstract: A dynamical system which is described in terms of an idempotent algebra by means of a vector equation with random irreducible matrix is considered. An approach based on approximation of the matrix of the system by means of matrices of simple structure is applied to evaluate bounds on the mean rate of growth of the state vector of the system. The process of constructing the approximations is reduced to the solution of problems of minimization of certain numerically valued functions. Examples that illustrate the evaluation of bounds on the mean rate of growth of the state vector for a system with matrix of dimension 2 are presented.
15 citations
TL;DR: In this paper, it was shown that if R is a ring with identity, and S and A op are the functor rings associated to the categories Mod(R) and Mod(A op ), respectively, then there is a duality between the categories of finitely presented objects of Mod(S op ) and Mod (A).
Abstract: It is known that if R is a ring with identity, and S and A op are the functor rings associated to the categories Mod(R) and Mod(R op ), respectively, then there is a duality between the categories of finitely presented objects of Mod(S op ) and Mod(A). We prove here this result in a more general case, namely when R is an idempotent ring, not necessarily having an identity, and when the categories Mod(R) of torsionfree and unitary right R-modules and Mod(R op ) of torsionfree and unitary left R-modules are locally finitely presented.
TL;DR: In this article, it was shown that there is a one to one correspondence between quantifiers and non-commutative binary operations making Q an idempotent and right-sided quantale.
Abstract: Let Q be an idempotent and right-sided quantale. There is a one to one correspondence between quantifiers and non-commutative binary operations making Q an idempotent and right-sided quantale. If Q is an atomic and irreducible orthomodular lattice there are only two such operations. Namely, the discrete quantifier and the indiscrete quantifer.
TL;DR: Wang et al. as mentioned in this paper showed that the additive reduct of a band semiring is a regular band and proved that the relation between the reduct and the band is not a congruence relation.
Abstract: The additive reduct of a band semiring is a regular band. We refer to [1] and [2] for a general background on semigroups, and on bands in particular. A semiring (S,+, ·) is said to be an idempotent semiring if both the reducts (S,+) and (S, ·) are bands. Green’s D [L-, R-] relation on the additive reduct (S,+) will be denoted by + D [ + L , + R ]. It is easy to see that if (S,+, ·) is an idempotent semiring, then + D is a semiring congruence, whereas + L induces a congruence on the multiplicative reduct (S, ·) (see also [3]). An idempotent semiring (S,+, ·) is said to be a band semiring if + D is the least (distributive) lattice congruence on S , or equivalently, if S satisfies the identities x+ xy + x ≈ x ≈ x+ yx+ x (see Theorem 2.2 of [3]). We shall show that the additive reduct of a band semiring is a regular band. We shall use the fact that a band is regular if and only if the Green’s relations L and R are congruence relations (see Section II.3 of [2]). Lemma. Let (S,+) be a band. The Green’s relation L is not a congrence relation on S if and only if S contains either a subband isomorphic to E1 or ∗The research is supported by an NSF(China) grant #10471112. The research of the first author is supported by a grant of the Youth Scintific Research Foundation of Southwest Normal University (#SWNUQ2004003). 440 Wang, Zhou, and Guo a subband isomorphic to E2 , where E1 and E2 are respectively given by + a b c d e a a a c d e b b b c d e c d e c d e d d d c d e e e e c d e and + a b c d e f g h a a a c d e c d e b b b f g h f g h c d e c d e c d e d d d c d e c d e e e e c d e c d e f g h f g h f g h g g g f g h f g h h h h f g h f g h . Proof. The Green’s relation L on the band S is not a congruence if and only if there exist a, b, k ∈ S such that aLb but k+a is not L-related to k+ b . If this is the case, then with c = a + k , the subband of S generated by a, b and c is isomorphic to E1 if c = b + c , and is isomorphic to E2 if c = b + c . Conversely, if S contains a subband isomorphic to E1 or E2 , then obviously the L-relation is not a congruence on S . Theorem. The additive reduct of a band semiring is a regular band. Proof. Let (S,+, ·) be a band semiring. Using the Lemma and duality, it suffices to show that the band (S,+) cannot contain a subband isomorphic to E1 or E2 . (1) Suppose that (S,+) has a subband T isomorphic to E1 . We identify T with E1 . First we have a = a+ ac+ a = a(a+ c+ a) = ad. Since a + Lb and + L is a congruence on (S, ·), we have a + Lab . Then a = ad = a(b+ d) = ab+ ad = ab+ a = ab. Since + D is a lattice congruence we have that ab + Dcb in S . So we have b = b+ db+ b = (b+ d+ b)b = db = (a+ c+ a)b = ab+ cb+ ab = ab = a, a contradiction. Therefore (S,+) has no subband isomorphic to E1 . (2) Suppose S has a subband T isomorphic to E2 and we identify T with E2 . First we have a = a+ ae+ a = a(a+ e+ a) = ae and b = b+ gb+ b = (b+ g + b)b = gb. Wang, Zhou, and Guo 441 Since + D is a lattice congruence, we obtain that ab + Ddb + Dah . So we have ab = ab+ db+ ab = (a+ d+ a)b = db and ab = ab+ ah+ ab = a(b+ h+ b) = ah. Thus, noticing that ac + Dcb + Dab , we have (c+ a)(c+ b) = (c+ a)c+ (c+ a)b = c+ ac+ cb+ ab = c+ (ac+ ab) = c+ a(c+ b) = c+ ae = c+ a = d and (c+ a)(c+ b) = c(c+ b) + a(c+ b) = c+ cb+ ac+ ab = c+ (cb+ ab) = c+ (c+ a)b = c+ db = c+ ab. Therefore we have c + ab = d . Similarly, applying the distributive law to (f+a)(f+b), one can get f+ab = h in combination with b = gb and ab = ah . Thus we obtain that d = c+ ab = (a+ f) + ab = a+ (f + ab) = a+ h = e, a contradiction. So S has no subband isomorphic to E2 either. Thus the proof is complete. Note 1. Let (S, ·) be a band and define an addition by: for a, b ∈ S, a = a + b . Then (S,+, ·) is a band semiring. Thus, any band is isomorphic to the multiplicative reduct of some band semiring. Note 2. The additive reduct of a band semiring need not be a normal band: see the example following Construction 3.5 of [3]. Note 3. For a band semiring S , the relation + L is a semiring congruence. From this and its dual it follows that S is a subdirect product of the semirings S/ + L and S/ + R . 442 Wang, Zhou, and Guo Acknowledgment The authors would like to thank Professor Francis Pastijn for his valuable suggestions and help. References [1] Howie, J. M., “Fundamentals of Semigroup Theory”, Clarendon Press, Ox- ford, 1995. [2] Petrich, M., “Lectures in Semigroups”, Akad. Berlag, Berlin, 1977. [3] Sen, M. K., Guo, Y. Q. and Shum, K. P., A class of idempotent semirings, Semigroup Forum 60 (2000), 351–367. Department of Mathematics Harbin Institute of Technology Harbin (150001), China Department of Mathematics Southwest China University Chongqing (400715), China zpwang@swu.edu.cn Department of Mathematics Jiangxi Normal University Nanchang (330027), China ylzhou185@163.com Department of Mathematics Southwest China University Chongqing (400715) China yqguo259@swu.edu.cn Received November 7, 2004 and in final form August 19, 2005 Online publication January 23, 2006
Book•
20 Oct 2005
TL;DR: In this article, the reduction to positive characteristic and completion of CG complements are discussed.Motivating Examples: Reduction to Positive Characteristic, Homological Approach, Completions of CG
Abstract: Motivating Examples.- Reduction to Positive Characteristic.- A Homological Approach.- Completions of CG.
TL;DR: In this paper, equational theories of several left symmetric left distributive operations on groups are studied and the normal forms of terms in the variety of LSLD groupoids, L SLD medial groupoid, LSLd idempotent groupoid and LSLDL medial idemetric groupoid are found.
Abstract: We study equational theories of several left symmetric left distributive operations on groups. Normal forms of terms in the variety of LSLD groupoids, LSLD medial groupoids, LSLD idempotent groupoids and LSLD medial idempotent groupoids are found.
TL;DR: In this article, the authors obtained the general form of bijective order and orthogonality preserving maps on the poset of all n-×-n upper triangular idempotent matrices over an arbitrary field F.
Journal Article•
TL;DR: In this article, rank egualities for idempotent and involutory matrices and necessary and sufficient conditions for a P+bQ bing invertible were given.
Abstract: This paper shows several rank egualities for idempotent and involutory matrices and gives several mecessary and sufficient conditions for aP+bQ bing invertible (P and Q are idempotent matrices);and gives several necessary and sufficient conditions for A+B+2I_n being invertible(A and B are involutory matrices).
TL;DR: The Klyachko idempotent has attracted interest from combinatorialists, partly because its denition involves the major index of permutations, and the proof that it is a Lie element emerges from Stanley’s theory of P -partitions.
Abstract: Because they play a role in our understanding of the symmetric group algebra, Lie idempotents have received considerable attention. The Klyachko idempotent has attracted interest from combinatorialists, partly because its denition involves the major index of permutations. For the symmetric group Sn, we look at the symmetric group algebra with coecients from the eld of rational functions in n variables q1;:::;qn .I n this setting, we can dene an n-parameter generalization of the Klyachko idempotent, and we show it is a Lie idempotent in the appropriate sense. Somewhat surprisingly, our proof that it is a Lie element emerges from Stanley’s theory of P -partitions.
TL;DR: In this article, it was shown that a mode (idempotent and entropic algebra) is a Lallement sum of its cancellative submodes over a normal band if it has a congruence with the normal band quotient and cancellative classes.
Abstract: A mode (idempotent and entropic algebra) is a Lallement sum of its cancellative submodes over a normal band if it has a congruence with a normal band quotient and cancellative congruence classes. We show that such a sum embeds as a subreduct into a semimodule over a certain ring, and discuss some consequences of this fact. The result generalizes a similar earlier result of the authors proved in the case when the normal band is a semilattice.
TL;DR: In this article, the authors considered the problem of generating a skew lattice in R that splits, having order dividing 16, given idempotents e and f in a ring R, if ef and fe are also idempotsent.
Abstract: Given idempotents e and f in a ring R, if ef and fe are also idempotent, then e and f generate a skew lattice in R that splits, having order dividing 16. The skew Boolean algebra generated from e and f is also considered.
TL;DR: -'a is a product of idempotent mappings in T by Theorem 1.1.1, demonstrating that a = 38.
Abstract: that some subset of B is mapped bijectively to a(B). But then f-' a is a mapping in TB. Moreover, ,-6 is noninvertible since a is noninvertible. Hence -'a is a product of idempotent mappings in T by Theorem 1.1. Each of these idempotent mappings can be extended to an idempotent linear mapping from V to V. Let 8 denote the product of these linear mappings. It follows that a(B) = 8(/--1)(B) = 8(B), demonstrating that a = 38. Because both 8 and /3 are products of idempotents, so is a. U
Book•
01 Jan 2005
TL;DR: The Maslov's dequantization, idempotent and tropical mathematics: A very brief introduction by G. L. Shpiz as discussed by the authors is a generalization of the utility theory using a hybrid IDEMP-probabilistic measure.
Abstract: The Maslov's dequantization, idempotent and tropical mathematics: A very brief introduction by G. L. Litvinov Set coverings and invertibility of functional Galois connections by M. Akian, S. Gaubert, and V. Kolokoltsov Discrete max-plus spectral theory by M. Akian, S. Gaubert, and C. Walsh Dequantization of coadjoint orbits: Moment sets and characteristic varieties by A. Baklouti On the combinatorial aspects of max-algebra by P. Butkovic Max-plus convex sets and functions by G. Cohen, S. Gaubert, J.-P. Quadrat, and I. Singer Algebras of Lukasiewicz's logic and their semiring reducts by A. Di Nola and B. Gerla Max-plus approaches to continuous space control and dynamic programming by W. H. Fleming and W. M. McEneaney A blow-up phenomenon in the Hamilton-Jacobi equation in an unbounded domain by K. Khanin, D. Khmelev, and A. Sobolevskii The dequantization transform and generalized Newton polytopes by G. L. Litvinov and G. B. Shpiz An object-oriented approach to idempotent analysis: Integral equations as optimal control problems by P. Loreti and M. Pedicini Traffic assignment & Gibbs-Maslov semirings by P. Lotito, J.-P. Quadrat, and E. Mancinelli Viscosity solutions on Lagrangian manifolds and connections with tunnelling operators by D. McCaffrey Applications of the generated pseudo-analysis to nonlinear partial differential equations by E. Pap A generalization of the utility theory using a hybrid idempotent-probabilistic measure by E. Pap Amoebas: Their spines and their contours by M. Passare and A. Tsikh First steps in tropical geometry by J. Richter-Gebert, B. Sturmfels, and T. Theobald On minimax and idempotent generalized weak solutions to the Hamilton-Jacobi equation by I. V. Roublev Dequantisation: Semi-direct sums of idempotent semimodules by E. Wagneur On (min,max,+)-inequalities by J. van der Woude and G. J. Olsder Solution of some max-separable optimization problems with inequality constraints by K. Zimmermann.
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TL;DR: In this paper, a contractive, idempotent, MASA bimodule map on B(H) whose range is not a ternary subalgebra of B (H) was constructed.
Abstract: We construct a contractive, idempotent, MASA bimodule map on B(H), whose range is not a ternary subalgebra of B(H). Our method uses a crossed-product to reduce the existence of such an idempotent map to an analogous problem about the ranges of idempotent maps that are equivariant with respect to a group action and Hamana's theory of G-injective envelopes.
TL;DR: In this article, it was shown that a clone C with five essentially binary operations is minimal if and only if C is a clone of a non-trivial affine space over GF(7).
Abstract: In this paper, among other results, we prove that a clone C with five essentially binary operations is minimal if and only if C is a clone of a non-trivial affine space over GF(7). This result is a product of systematic investigation of varieties of idempotent commutative groupoids. It is probably the most spectacular result of this investigation, but many other results formulated and proved throughout the paper may turn out more useful in further study in this area.
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TL;DR: In this paper, the authors studied the closed algebra B_I(G) generated by the idempotents in the Fourier-Stieltjes algebra of a locally compact group G and showed that it is a regular Banach algebra with computable spectrum G^I.
Abstract: We study the closed algebra B_I(G) generated by the idempotents in the Fourier-Stieltjes algebra of a locally compact group G. We show that it is a regular Banach algebra with computable spectrum G^I, which we call the idempotent compactification of G. For any locally compact groups G and H, we show that B_I(G) is completely isometrically isomorphic to B_I(H) exactly when G/G_e= H/H_e, where G_e and H_e are the connected components of the identities. We compute some examples to illustrate out results.
Journal Article•
TL;DR: The main result is that the class of o-t-ts transformations computed by x bottom-up tree series transducers over A is incomparable (with respect to set inclusion) with the classes of e-tree-to- tree-series and o- Tree-To-Tree-series transducers computed by such transducers.
Abstract: Polynomial bottom-up and top-down tree series transducers over partially ordered semirings are considered, and the classes of e-tree-to-tree-series (for short: e-t-ts) and o-tree-to-tree-series (for short: o-t-ts) transformations computed by such transducers are compared. The main result is the following. Let A be a weakly growing semiring and x, y ∈ {deterministic, homomorphism}. The class of o-t-ts transformations computed by x bottom-up tree series transducers over A is incomparable (with respect to set inclusion) with the class of e-t-ts transformations computed by y bottom-up tree series transducers over A. Moreover, the latter class is incomparable with the class of e-t-ts transformations computed by x top-down tree series transducers over A. If additionally A is additively idempotent, then the above statements even hold for every x, y ∈ {polynomial, deterministic, homomorphism}.
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TL;DR: The first explicit description of the Baxter algebra on one generator is due to Ebrahimi-Fard and Guo as discussed by the authors, who provided an alternative description in terms of a certain class of trees, which form a linear basis for this algebra.
Abstract: We study the free (associative, non-commutative) Baxter algebra on one generator. The first explicit description of this object is due to Ebrahimi-Fard and Guo. We provide an alternative description in terms of a certain class of trees, which form a linear basis for this algebra. We use this to treat other related cases, particularly that in which the Baxter map is required to be quasi-idempotent, in a unified manner. Each case corresponds to a different class of trees.
Our main focus is on the underlying combinatorics. In several cases, we provide bijections between our various classes of trees and more familiar combinatorial objects including certain Schroeder paths and Motzkin paths. We calculate the dimensions of the homogeneous components of these algebras (with respect to a bidegree related to the number of nodes and the number of angles in the trees) and the corresponding generating series. An important feature is that the combinatorics is captured by the idempotent case; the others are obtained from this case by various binomial transforms. We also relate free Baxter algebras to Loday's dendriform trialgebras and dialgebras. We show that the free dendriform trialgebra (respectively, dialgebra) on one generator embeds in the free Baxter algebra with a quasi-idempotent map (respectively, with a quasi-idempotent map and an idempotent generator). This refines results of Ebrahimi-Fard and Guo.
01 Jan 2005
TL;DR: In this article, the uniqueness of regular expressions is recognized modulo associativity, commutativity, and idempotence of the union operator, and the number of derivatives can be further reduced through simplification based on the identities for regular expressions.
Abstract: Derivatives of regular expressions were first introduced by Brzozowski in [1]. By recursively computing all derivatives of a regular expression, and associating a state with each unique derivative, a deterministic finite automaton can be constructed. Convergence of this process is guaranteed if uniqueness of regular expressions is recognized modulo associativity, commutativity, and idempotence of the union operator. Additionaly, through simplification based on the identities for regular expressions, the number of derivatives can be further reduced.
Journal Article•
TL;DR: Characterize the mutiplicative band semirings in different ways to research the suddirect product decomposition of the idempotent semiring in R·D.
Abstract: Aim To study a very important class of idempotent semiring,so-called multiplicative band semirings.Methods Characterize the mutiplicative band semirings in different ways.Results The characters is given on of the idempotent semiring in R·D research the suddirect product decomposition.Conclusion Semirings S belongs to R·D if and only if Green′s D-relation D· on the mutiplication reduct(S,·) of S is lattice Congruence on S;R·D=L·_zD∨R·_zD.
Journal Article•
TL;DR: In this article, the structure of the pseudo-strong right normal idempotent semiring of C-semirings was constructed, and it was shown that an A-idempotent semiiring is a normal semiring, if and only if it is a pseudo strong right semiiring of left zero semirings.
Abstract: In this paper,we construct the structure of the pseudo-strong right normal idempotent semiring of C-semirings,and then an A-idempotent semiring is a normal idempotent semiring,if and only if it is a pseudo-strong right (normal) idempotent semiring of left zero semirings.This result last gets the characterization of the direct product of left normal A-idempotent semiring and a ring as a pseudo-strong semilattice idempotent semiring of left rings,and some corollaries.