Showing papers on "Idempotence published in 1999"
TL;DR: In this article, the authors give conditions under which an idempotent measure has a density and show by many examples that they are often satisfied, depending on the lattice structure of the semiring and on the Boolean algebra in which the measure is defined.
Abstract: Considering measure theory in which the semifield of positive real numbers is replaced by an idempotent semiring leads to the notion of idempotent measure introduced by Maslov. Then, idempotent measures or integrals with density correspond to supremums of functions for the partial order relation induced by the idempotent structure. In this paper, we give conditions under which an idempotent measure has a density and show by many examples that they are often satisfied. These conditions depend on the lattice structure of the semiring and on the Boolean algebra in which the measure is defined. As an application, we obtain a necessary and sufficient condition for a family of probabilities to satisfy the large deviation principle as defined by Varadhan.
119 citations
TL;DR: It is shown that sup- and inf-decomposable measures can be obtained as limits of families of Pseudo-additive measures with respect to generated pseudo-additions as well as corresponding integrals of g-integrals.
94 citations
TL;DR: It is proved that BZMV dM - algebras (which are equationally characterized) are the same as the Stonian MV- algeBRas and a rst representation theorem is proved.
42 citations
TL;DR: In this paper, the authors studied idempotent analogs of topological tensor products in the sense of A. Grothendieck, and the basic concepts and results are expressed in purely algebraic terms.
Abstract: We study idempotent analogs of topological tensor products in the sense of A. Grothendieck. The basic concepts and results are expressed in purely algebraic terms. This is one of a series of papers on idempotent functional analysis.
32 citations
TL;DR: It is proved that the recognizable series are certain rational power series, which can be constructed from the polynomials by using the operations sum, product, and a restricted star which is applied only to series for which the elements in the support all have the same connected alphabet.
Abstract: Kleene's theorem on the coincidence of regular and rational languages in free monoids has been generalized by Schutzenberger to a description of the recognizable formal power series in noncommuting variables over arbitrary semirings and by Ochmanski to a characterization of the recognizable languages in trace monoids. We will describe the recognizable formal power series over arbitrary semirings and in partially commuting variables, i.e. over trace monoids. We prove that the recognizable series are certain rational power series, which can be constructed from the polynomials by using the operations sum, product, and a restricted star which is applied only to series for which the elements in the support all have the same connected alphabet. The converse is true if the underlying semiring is commutative. Moreover, if in addition the semiring is idempotent then the same result holds with a star restricted to series for which the elements in the support have connected (possibly different) alphabets. It is shown that these assumptions over the semiring are necessary. This provides a joint generalization of Kleene's, Schutzenberger's and Ochmanski's theorems.
29 citations
TL;DR: In this article, a solution for the word problem for free idempotent distributive semirings is given, and the latticeL (ID) of subvarieties of the variety ID of all band varieties is determined.
Abstract: A solution is given for the word problem for free idempotent distributive semirings. Using this solution the latticeL (ID) of subvarieties of the variety ID of idempotent distributive semirings is determined. It turns out thatL (ID) is isomorphic to the direct product of a four-element lattice and a lattice which is itself a subdirect product of four copies of the latticeL(B) of all band varieties. ThereforeL(ID) is countably infinite and distributive. Every subvariety of ID is finitely based.
22 citations
TL;DR: It is shown that the free model of the new set of axioms is a class of trees labelled over A, and the three proposed interpretations of regular expressions (algebraic, denotational, and behavioural) are proven to coincide.
19 citations
Posted Content•
TL;DR: In this article, a brief introduction to idempotent mathematics and an ideme-potent version of Interval Analysis is presented, and some applications are discussed, as well as a brief discussion of the application of idempots.
Abstract: A brief introduction into Idempotent Mathematics and an idempotent version of Interval Analysis are presented. Some applications are discussed.
17 citations
TL;DR: In this article, a simple property complementary to idempotentity is defined: a projection is idem-potent and a linear operator is a projection, which is useful in the theory of non-linear smoothers.
Abstract: Projections are important in many fields of analysis, specifically, in approximation theory and signal analysis. They can be perceived to separate a vector (function, sequence) into a component in a chosen specific subspace and the residue, or the “signal” and “noise”. In the framework of linear theory a projection is idem-potent and an idempotent linear operator is a projection. When an operator is not linear the ideas are still partially applicable and useful. A simple property, complementary to idempotence, extends the analogy in a useful way. This is particularly so in the theory of non-linear smoothers, when selectors are analysed and compared in a LULU-structure.
15 citations
TL;DR: The trilattice algebra as mentioned in this paper is a triadic generalization of lattices, which can be seen as an algebraic approach to the triadic construction of lattice.
Abstract: This paper presents the new algebra of trilattices, which are understood as the triadic generalization of lattices. As with lattices, there is an order-theoretic and an algebraic approach to trilattices. Order-theoretically, a trilattice is defined as a triordered set in which six triadic operations of some small arity exist. The Reduction Theorem guarantees that then also all finitary operations exist in trilattices. Algebraically, trilattices can be characterized by nine types of trilattice equations. Apart from the idempotent, associative, and commutative laws, further types of identities are needed such as bounds and limits laws, antiordinal, absorption, and separation laws. The similarities and differences between ordered and triordered sets, lattices and trilattices are discussed and illustrated by examples.
12 citations
TL;DR: An alternative (tree-based) semantics for a class of regular expressions is proposed that assigns a central role to the + operator and thus to nondeterminism and nondeterministic choice.
Abstract: An alternative (tree-based) semantics for a class of regular expressions is proposed that assigns a central role to the + operator and thus to nondeterminism and nondeterministic choice. For the new semantics a consistent and complete axiomatization is obtained from the original axiomatization of regular expressions by Salomaa and by Kozen by dropping the idempotence law for + and the distribution law of • over +.
TL;DR: In this paper, the authors used the structural theorem proved in [R3] to describe all subdirectly irreducible groupoids in each nontrivial subvariety of the variety SIE of all SIE-groupoids.
Abstract: This paper is a sequel to the author's work [R3]. It is concerned with the representation of SIE-groupoids as subdirect products. We will use the structural theorem proved in [R3] to describe all subdirectly irreducible groupoids in each nontrivial subvariety of the variety SIE of all SIE-groupoids. The notation and terminology of [R3] will be used without explanation or apology in this paper. From now on let us assume that a subset {gi \\ i £ J} of elements of G generates the SIE-groupoid (G, •), and denote by Gi the orbit of the generator gi, i.e. Gi — {g € G \\ g = gib\\...bk, for some elements b\\,...,bic from G}. Let w denote the congruence relation on (G, •), defined by [R3, (3.1)], decomposing it into the disjoint sum of orbits Gi, for i £ I C J. On each orbit we have defined (see [R3, (3.5)]) an abelian group structure (Gi,+,gi). It was proved in [R3] that the SIE-groupoid (G, •) is the AG-sum of the abelian groups (Gi, +), for i € / , by the mappings hj : Gi Gj, a i—• gjd.
TL;DR: In this article, the relation between localizations and completions was studied using the machinery of orthogonal pairs, and general properties of idempotent approximations to monads were given.
TL;DR: In this article, the structure of implicit operations on R ∩ LJ1, the pseudovariety of all R-trivial, locally idempotent and locally commutative semigroups, is studied.
Abstract: This paper is concerned with the structure of implicit operations on R ∩ LJ1, the pseudovariety of all R-trivial, locally idempotent and locally commutative semigroups. We give a unique factorization statement, in terms of component projections and idempotent elements, for the implicit operations on R ∩ LJ1. As an application we give a combinatorial description of the languages that are both R-trivial and locally testable. A similar study is conducted for the pseudovariety DA ∩ LJ1 of locally idempotent and locally commutative semigroups in which each regular D-class is a rectangular band.
TL;DR: A general notion of pseudo-convolution based on pseudo-arithmetical operations is recalled and the idempotents with respect to pseudo-convolutions are investigated.
TL;DR: In this article, the question of whether the sets of idempotents in weakly almost periodic compacti cations of (N, + +) and (Z, +) are closed is answered negatively.
Abstract: This paper answers negatively the question of whether the sets
of idempotents in the weakly almost periodic compacti�cations of (N; +) and
(Z; +) are closed.
Journal Article•
01 Jan 1999
TL;DR: A semigroup (M,*) consists of a nonempty set M on which an associative operation * is defined as discussed by the authors, and a semigroup can be canonically embedded in a monoid M ∪ {e} where e is some element not in M. This element can easily seen to be unique.
Abstract: A semigroup (M, *) consists of a nonempty set M on which an associative operation * is defined. If M is a semigroup in which there exists an element e satisfying m * e = m = e * m for all m ∈ M, then M is called a monoid having identity element e. This element can easily seen to be unique, and is usually denoted by 1m- Note that a semigroup (M, *) which is not a monoid can be canonically embedded in a monoid M′ - M ∪ {e} where e is some element not in M, and where the operation * is extended to an operation on M′ by defining e * M′ = M′ = M′ * e for all m′ ∈ m′. An element m of M idempotent if and only if m * m = m. A semigroup (M, *) is commutative if and only if m * M′ = M′ * m for all m.m′ ∈ M.
01 Jan 1999
TL;DR: In this paper, the problem of finding a pattern equation with a solution in a free idempotent semigroup is shown to be NP-complete, where X is a sequence of variables and A is a series of constants.
Abstract: A pattern equation is a word equation of the form X = A where X is a sequence of variables andA is a sequence of constants. The problem whether X = A has a solution in a free idempotent semigroup (free band) is shown to be NP-complete.
01 Jan 1999
TL;DR: In this article, it was shown that a commutative idempotent groupoid is a proper Plonka sum of affine spaces over GF(3) if and only if it has 27 essentially 4-ary term functions.
Abstract: We prove in this paper that a commutative idempotent groupoid is a proper Plonka sum of affine spaces over GF(3) if and only if it has 27 essentially 4-ary term functions. This means, in another terminology, that the number 27 is the characteristic number of those sums in the variety of all commutative idempotent groupoids. Using this result, we also show that a medial idempotent groupoid with three essentially binary operations contains a subgroupoid term equivalent to either the five-element affine space over GF(5) or a Plonka sum of a trivial groupoid and the three-element affine space over GF(3).
TL;DR: In this article, the authors apply the theory of inductive groupoids, in particular the construction of the idempotent generated regular semigroup given in §6 of [8] to detemine some combinatorial properties of the semigroup Sn.
Abstract: be a vector space of dimension n over a field K. Here we denote by Sn the set of all singular endomorphisms of V. Erdos [5], Dawlings [4] and Thomas J. Laffey [6] have shown that Sn is an idempotent generated regular semigroup. In this paper we apply the theory of inductive groupoids, in particular the construction of the idempotent generated regular semigroup given in §6 of [8] to detemine some combinatorial properties of the semigroup Sn.
Proceedings Article•
01 Jan 1999
TL;DR: In this article, all possible idempotent operators on a finite chain L which are associative, commutative and non-decreasing in each place are characterized.
Abstract: This work is devoted to find and study some possible idempotent operators on a finite chain L. Specially, all idempotent operators on L which are associative, commutative and non-decreasing in each place are characterized. By adding one smoothness condition, all these operators reduce to special combinations of Minimum and Maximum.
TL;DR: In this paper, it was shown that Green's Indecomposability Theoremi fails for infinitely generated modules by applying some general properties of idempotent modules, and they were also able to construct explicit examples of modules for which the cancellation property fails.
Abstract: Almost all of the basic theorems in the representation theory of finite groups have proofs that depend upon the Krull-Schmnidt Theorem Because this theorem holds only for finite-dimensional modules, however, the recent interest in infinitely generated modules raises the question of which results may hold more generally In this paper we present an example showing that Green's Indecomposability Theoremi fails for infinitely generated modules By developing and applying some general properties of idempotent modules, we are also able to construct explicit examples of modules for which the cancellation property fails
TL;DR: In this paper, it was shown that p-subspaces obtained from the Peirce decomposition of a Bernstein algebra A relative to an idempotent have dimensions which are independent of the ideme-potent used to decompose A.
Abstract: The purpose of this paper is to prove that some vector subspaces, called p-subspaces, obtained from the Peirce decomposition of a Bernstein algebra A relative to an idempotent have dimensions which are independent of the idempotent used to decompose A. In particular, for Bernstein-Jordan algebras, this fact is true for every such subspace and this implies that all p-subspaces of a Bernstein algebra, contained in V, for A = Ke + U + V, have invariant dimension. Finally we classify all p-subspaces of degree ≥ 3, contained in U, in a Bernstein algebra A, relative to the invariance (or not) of dimension.
TL;DR: Different approaches will be shown: algebraic properties, one side invertibility and idempotency, and certain subsets of proper combinators and Church algebras between them will be proved to be domains consisting only of fixed points of combinators.
Abstract: Solutions of the equations X = ZX and X = XZ are found and discussed for Z, X normal terms of the lambda-calculus. Obviously fixed point combinators are of no help. Solutions will be independent from any kind of godelization or coding of data structures, they will be provided by typeless self-application. Different approaches will be shown: algebraic properties, one side invertibility and idempotency. Certain subsets of proper combinators and Church algebras between them will be proved to be domains consisting only of fixed points of combinators.
Journal Article•
TL;DR: In this paper, it was shown that if the torsion subloop of an RA loop L is central, then there exist no nonzero nilpotent elements in the loop algebra over any commutative ring, with some obvious exceptions.
Abstract: We show that if the torsion subloop of an RA loop L is central, then there exist no nonzero nilpotent elements in the loop algebra over any commutative ring, with some obvious exceptions. We also study the nilpotency of augmentation ideals and prove that integral loop rings contain no nontrivial idempotent ideals.
Journal Article•
TL;DR: The sign patterns of symmetric idempotent matrices through order 5 are determined and various constructions are presented.
Abstract: Sign patterns of idempotent matrices, especially symmetric idempotent matrices, are investigated. A number of fundamental results are given and various constructions are presented. The sign patterns of symmetric idempotent matrices through order 5 are determined. Some open questions are also given.