Showing papers on "Idempotence published in 2001"
TL;DR: In this paper, an algebraic approach to idempotent functional analysis is presented, which is an abstract version of the traditional functional analysis developed by V. P. Maslov and his collaborators.
Abstract: This paper is devoted to Idempotent Functional Analysis, which is an “abstract” version of Idempotent Analysis developed by V. P. Maslov and his collaborators. We give a brief survey of the basic ideas of Idempotent Analysis. The correspondence between concepts and theorems of traditional Functional Analysis and its idempotent version is discussed in the spirit of N. Bohr's correspondence principle in quantum theory. We present an algebraic approach to Idempotent Functional Analysis. Basic notions and results are formulated in algebraic terms; the essential point is that the operation of idempotent addition can be defined for arbitrary infinite sets of summands. We study idempotent analogs of the basic principles of linear functional analysis and results on the general form of a linear functional and scalar products in idempotent spaces.
222 citations
Book•
07 May 2001
TL;DR: In this paper, the Laplace-Fenchel model is used to transform Idempotent Probability Measures on topological spaces into Projective Limits and Maxingales Stopping Times.
Abstract: IDEMPOTENT PROBABILITY THEORY Idempotent Probability Measures Idempotent Measures Measurable Maps Modes of Convergence Idempotent Integration Product Spaces Independence and Conditioning Idempotent Distributions and Laplace-Fenchel Transforms' Idempotent Measures on Topological Spaces Idemptent Measures on Projective Limits Topological Spaces of Idempotent Probabilities Maxingales Stopping Times Idempotent Stochastic Processes Exponential Maxingales Wiener and Poisson Idempotent Processes Continuous Local Maxingales Idempotent Ito Equations Semimaxingales and Maxingale Problems Proofs of the Uniqueness Results Convergence of Idempotent Processes LARGE DEVIATION CONVERGENCE Large Deviation Convergence in Tihonov Spaces General Theory Large Deviation Convergence in the Skorohod Space The Method of Finite-Dimensional Distributions Convergence of Stochastic Exponentials LD Convergence via Convergence of the Characteristics Corollaries The Method of the Maxingale Problem Convergence of Stochastic Exponentials Convergence of Characteristics APPLICATIONS Markov Processes Queueing Networks APPENDIX
143 citations
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TL;DR: The correspondence principle is used to develop an approach to object-oriented software and hardware design for algorithms of idempotent calculus.
Abstract: This paper is devoted to heuristic aspects of the so-called idempotent calculus There is a correspondence between important, useful and interesting constructions and results over the field of real (or complex) numbers and similar constructions and results over idempotent semirings in the spirit of N Bohr's correspondence principle in Quantum Mechanics
Some problems nonlinear in the traditional sense (for example, the Bellman equation and its generalizations) turn out to be linear over a suitable semiring; this linearity considerably simplifies the explicit construction of solutions
The theory is well advanced and includes, in particular, new integration theory, new linear algebra, spectral theory and functional analysis It has a wide range of applications
Besides a survey of the subject, in this paper the correspondence principle is used to develop an approach to object-oriented software and hardware design for algorithms of idempotent calculus
125 citations
TL;DR: In this article, interval analysis over idempotent semirings is applied to construction of exact interval solutions to the interval discrete stationary Bellman equation, which is typically NP-hard in the traditional interval linear algebra.
Abstract: Many problems in optimization theory are strongly nonlinear in the traditional sense but possess a hidden linear structure over suitable idempotent semirings. After an overview of "Idempotent Mathematics" with an emphasis on matrix theory, interval analysis over idempotent semirings is developed. The theory is applied to construction of exact interval solutions to the interval discrete stationary Bellman equation. Solution of an interval system is typically NP-hard in the traditional interval linear algebra; in the idempotent case it is polynomial. A generalization to the case of positive semirings is outlined.
77 citations
27 Aug 2001
TL;DR: Pin's refinement of Eilenberg theorem gives a one-to-one correspondence between positive varieties of rational languages and pseudovarieties of ordered monoids.
Abstract: A classical construction assigns to any language its (ordered) syntactic monoid. Recently the author defined the so-called syntactic semiring of a language. We discuss here the relationships between those two structures. Pin's refinement of Eilenberg theorem gives a one-to-one correspondence between positive varieties of rational languages and pseudovarieties of ordered monoids. The author's modification uses so-called conjunctive varieties of rational languages and pseudovarieties of idempotent semirings. We present here also several examples of our varieties of languages.
45 citations
TL;DR: In this paper, the continuity of range projections of idempotents in C*-algebras is analyzed, and a Schur type decomposition is obtained, which leads to simple proofs of results on Moore-Penrose inverse and norms.
Abstract: In this paper we study range projections of idempotents in C*-algebras, and use them to obtain a Schur type decomposition that leads to simple proofs of results on Moore-Penrose inverse and norms of idempotents. We analyze the continuity of range projections, obtain a general result on their approximation, and recover a result of Vidav on two projections in a Hilbert space. Several representations of range projections are given. 1. Range projections Basic facts about C*-algebras needed in this paper can be found, for example, in Davidson's monograph [3]. In this paper, 21 is a unital C*algebra with unit I . By 2 l 1 we denote the set of all invertible elements of 21. We recall that I + A*A € 21\" for all A € 21 and that \\\\A*A\\\\ = \\\\A\\\\ (the C*-identity). An element A € 21 is polar if 0 is at most a pole of the resolvent of A, and quasipolar if 0 is an isolated singularity of the resolvent of A. We will need the following characterization of quasipolar elements of 21. LEMMA 1.1 [9, Theorem 4.2]. An element A G 21 is quasipolar if and only if there exists an idempotent P € 21 commuting with A such that AP is quasinilpotent and A + P E 2 l 1 . Such idempotent is unique, and is called the spectral idempotent of A corresponding to 0, written A. From the preceding lemma it follows that A is polar if and only it is quasipolar and AA = 0 for some integer k. In particular, A is simply polar if and only if A is quasipolar and AA = 0. The word 'projection' will be reserved for an element Q of a C*-algebra 21 which is self-adjoint and idempotent, that is, Q* = Q = Q. A motivation for 1991 Mathematics Subject Classification: 46L05, 46H30, 47A60.
41 citations
TL;DR: In this article, the main facts of idempotent analysis and its major areas of applications are reviewed, including multicriteria optimisation, turnpike theory and mathematical economics, in the theory of generalised solutions.
Abstract: Consider the set A = R ∪ {+∞} with the binary operations o1 = max
and o2 = + and denote by An the set of vectors v = (v1,,vn) with entries
in A Let the generalised sum u o1 v of two vectors denote the vector with
entries uj o1 vj , and the product a o2 v of an element a ∈ A and a vector
v ∈ An denote the vector with the entries a o2 vj With these operations,
the set An provides the simplest example of an idempotent semimodule
The study of idempotent semimodules and their morphisms is the subject
of idempotent linear algebra, which has been developing for about
40 years already as a useful tool in a number of problems of discrete optimisation
Idempotent analysis studies infinite dimensional idempotent
semimodules and is aimed at the applications to the optimisations problems
with general (not necessarily finite) state spaces We review here
the main facts of idempotent analysis and its major areas of applications
in optimisation theory, namely in multicriteria optimisation, in turnpike
theory and mathematical economics, in the theory of generalised solutions
of the Hamilton-Jacobi Bellman (HJB) equation, in the theory of games
and controlled Marcov processes, in financial mathematics
39 citations
TL;DR: In this article, a new type of elements of n-ary (n ≥ 3) semigroups with an idempotent identity was selected and properties of ideals connected with these elements were investigated.
Abstract: We select a new type of elements of n-ary (n ≥ 3) semigroups with an idempotent and investigate properties of ideals connected with these elements.
37 citations
TL;DR: In this article, the structure of semigroups of implicit operations on the pseudovariety L Sl of finite locally idempotent and locally commutative semigroup is studied.
25 citations
TL;DR: It is proved that the set of all fuzzy congruences on a regular semigroup contained in δ H forms a modular lattice, where δH is the characteristic function of H and H is the H -equivalent relation on the semigroup.
23 citations
TL;DR: In this paper, the authors show that certain varieties of idempotent semirings are determined by some properties of Green's relations, provide equational bases for them and give conditions guaranteeing that some Green relations are congruences.
Abstract: We show that certain varieties of idempotent semirings are determined by some properties of Green's relations, provide equational bases for them and give conditions guaranteeing that some Green's relations are congruences.
TL;DR: In this article, it was shown that the ring of polynomials in one indeterminate over a nil ring cannot be homomorphically mapped onto a ring containing a nonzero idempotent.
TL;DR: A polynomial time algorithm is given to decide whether or not a variety generated by an order-primal algebra admits a near unanimity function and so the problem of Larose and Zádori is answered.
Abstract: In this paper we introduce a new version of the concept of order varieties. Namely, in addition to closure under retracts and products we require that the class of posets should be closed under taking idempotent subalgebras. As an application we prove that the variety generated by an order-primal algebra on a finite connected poset P is congruence modular if and only if every idempotent subalgebra of P is connected. We give a polynomial time algorithm to decide whether or not a variety generated by an order-primal algebra admits a near unanimity function and so we answer a problem of Larose and Zadori.
01 Jan 2001
TL;DR: In this article, a semicentral reduced ring is defined as a ring which has right FPF, right nonsingular, or left perfect idempotents in R.
Abstract: An idempotent e of an algebra R is left semicentral if Re = eRe. If 0 and 1 are the only left semicentral idempotents in R, then R is called semicentral reduced. Recent results on generalized triangular matrix algebras and semicentral reduced algebras are surveyed. New results are provided for endomorphism algebras of modules and for semicentral reduced algebras. In particular, semicentral reduced rings which are right FPF, right nonsingular, or left perfect are described.
TL;DR: For a suitable series of idempotent ideals, a method of constructing tilting modules of finite projective dimension is given in this article, where the tilting module is constructed for a suitable set of ideals.
TL;DR: The main result in this article is that f ∈ L (R ) if and only if there exist an invertible matrix U ∈ T n (R) and an idempotence e ∈ R such that f(X)=U(eX+(1−e)X δ )U −1 for any X=(x ij )∈T n ( R ), where X δ =(x n+1−j n+ 1−i ).
TL;DR: In this article, a family of fuzzy measures is associated with each aggregation operator, and the properties of horizontal or vertical pseudo-additivity are recognizable by means of this family of fuzz measures.
Abstract: The natural properties of the aggregation operators and the most elementary ones are the idempotence, the monotonicity and the continuity from below. We assume only these properties for the aggregation operators with infinitely many inputs, defined by functionals on the family of measurable functions. A family of fuzzy measures is associated with each aggregation operator. The properties of horizontal or vertical pseudo-additivity are recognizable by means of this family of fuzzy measures.
TL;DR: It is proved that in Sh(Q), a finitely generated projective module is free (Theorem 7.2), a result that is relevant to the study of representation of non-commutative C∗-algebras.
TL;DR: The authors showed that the validity of Parikh's theorem for context-free languages depends only on a few equational properties of least pre-fixed points, and they showed an infinite basis of mu-term equations of continuous commutative idempotent semirings.
Abstract: We show that the validity of Parikh's theorem for context-free languages depends only on a few equational properties of least pre-fixed points. Moreover, we exhibit an infinite basis of mu-term equations of continuous commutative idempotent semirings.
TL;DR: In this article, the singular rank and the idempotent rank of subsemigroups of the full transformation semigroup that contain all singular transformations were derived for singular transformations.
Abstract: We compute the singular rank and the idempotent rank of those subsemigroups of the full transformation semigroup that contain all singular transformations.
TL;DR: Finite posets admitting an n-ary idempotent totally symmetric operation for all n are characterized in terms of zigzags, special objects related to the poset.
Abstract: An n-ary operation f is totally symmetric if it obeys the identity f(x1,...,xn)=f(y1,...,yn) for all sets of variables such that {x1,...,xn}={y1,...,yn}. We characterize finite posets admitting an n-ary idempotent totally symmetric operation for all n. The characterization is expressed in terms of zigzags, special objects related to the poset. Some open problems concerning idempotent Malcev conditions for order primal algebras are mentioned.
TL;DR: The main purpose is to obtain the inequalities involving the cosines of E, and to show that equality is closely related to Γ being Q-polynomial with respect to E, which generalizes a result of Lang on bipartite graphs and a results of Pascasio on tight graphs.
Journal Article•
TL;DR: In this article, Pin's refinement of Eilenberg theorem gives a one-to-one correspondence between positive varieties of rational languages and pseudovarieties of ordered monoids.
Abstract: A classical construction assigns to any language its (ordered) syntactic monoid. Recently the author defined the so-called syntactic semiring of a language. We discuss here the relationships between those two structures. Pin's refinement of Eilenberg theorem gives a one-to-one correspondence between positive varieties of rational languages and pseudovarieties of ordered monoids. The author's modification uses so-called conjunctive varieties of rational languages and pseudovarieties of idempotent semirings. We present here also several examples of our varieties of languages.
TL;DR: In this article, general duality results for complete semimodules over idempotent semirings, like the max-plus semiring, were established for the case of complete semi-modules.
TL;DR: In this article, the authors construct the first example of a just non-finitely based variety of big groups, i.e., a variety which is not necessarily finitely based but all the proper subvarieties are finitely-based.
Abstract: A bigroup is a pair (H, π) consisting of a group H and an idempotent endomorphism π of H. One can consider π as a unary operation on H so a bigroup is a universal algebra. The aim of our paper is to construct the first example of a just non-finitely based variety of bigroups i.e. a variety which is non-finitely based but all whose proper subvarieties are finitely based. There is a close similarity between varieties of bigroups and varieties of groups so we hope that our result could help to construct a just non-finitely based variety of groups. *The first author was supported by NSERC, Canada.
01 Jul 2001
TL;DR: It is demonstrated how the concepts of algebraic representability and strongly-local reductions developed here and in [20] can be used to characterize the computational complexity/efficient approximability of a number of basic problems and their variants, on various abstract algebraic structures F.
Abstract: We demonstrate how the concepts of algebraic representability and strongly-local reductions developed here and in [20] can be used to characterize the computational complexity/efficient approximability of a number of basic problems and their variants, on various abstract algebraic structures F. These problems include the following: Algebra: Determine the solvability, unique solvability, number of solutions, etc., of a system of equations on F. Determine the equivalence of two formulas or straight-line programs on F.Optimization: Let ∈ > 0.Determine the maximum number of simultaneously satisfiable equations in a system of equations on F; or approximate this number within a multiplicative factor or n∈. Determine the maximum value of an objective function subject to satisfiable algebraically-expressed constraints on F; or approximate this maximum value within a multiplicative factor of n∈Given a formula or straight-line program, find a minimum size equivalent formula or straight-line program; or find an equivalent formula or straight-line program of size l f(minimum). Both finite and infinite algebraic structures are considered. These finite structures include all finite non-degenerate lattices and all finite rings or semi-rings with a nonzero element idempotent under multiplication (e.g. all non-degenerate finite unitary rings or semi-rings); and these infinite structures include the natural numbers, integers, real numbers, various algebras on these structures, all ordered rings, many cancellative semi-rings, and all infinite lattices with two elements a,b such that a is covered by b.
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TL;DR: In this article, interval analysis over idempotent semirings is applied to construction of exact interval solutions to the interval discrete stationary Bellman equation, which is typically NP-hard in the traditional interval linear algebra.
Abstract: Many problems in optimization theory are strongly nonlinear in the traditional sense but possess a hidden linear structure over suitable idempotent semirings. After an overview of `Idempotent Mathematics' with an emphasis on matrix theory, interval analysis over idempotent semirings is developed. The theory is applied to construction of exact interval solutions to the interval discrete stationary Bellman equation. Solution of an interval system is typically NP-hard in the traditional interval linear algebra; in the idempotent case it is polynomial. A generalization to the case of positive semirings is outlined.
Posted Content•
TL;DR: In this paper, a generalization of the abstraction function for Sharing that can be applied to any language, with or without the occurs-check, is presented, and results for soundness, idempotence and commutativity for abstract unification using this abstraction function are proven.
Abstract: It is important that practical data-flow analyzers are backed by reliably proven theoretical results. Abstract interpretation provides a sound mathematical framework and necessary generic properties for an abstract domain to be well-defined and sound with respect to the concrete semantics. In logic programming, the abstract domain Sharing is a standard choice for sharing analysis for both practical work and further theoretical study. In spite of this, we found that there were no satisfactory proofs for the key properties of commutativity and idempotence that are essential for Sharing to be well-defined and that published statements of the soundness of Sharing assume the occurs-check. This paper provides a generalization of the abstraction function for Sharing that can be applied to any language, with or without the occurs-check. Results for soundness, idempotence and commutativity for abstract unification using this abstraction function are proven.