Showing papers on "Idempotence published in 2002"

Associativity of crossed products by partial actions, enveloping actions and partial representations

Abstract: Given a partial action \alpha of a group G on an associative algebra A we consider the crossed product A x_\alpha G. Using the algebras of multipliers of ideals of A we prove that A x_\alpha G is associative, provided that all ideals of A are idempotent. This generalizes a previous result on the associativity of A x_\alpha G in the context of C*-algebras. We also give a criteria for the existence of a global extension of a given partial action on an algebra and use crossed products to study relations between partial actions of groups on algebras and partial representations. As an application we endow partial group algebras with crossed product structure.

Information Algebras: Generic Structures For Inference

Abstract: 1 Introduction.- 2 Valuation Algebras.- 2.1 The Framework.- 2.2 Axioms.- 2.3 Examples of Valuation Algebras.- 2.3.1 Indicator Functions.- 2.3.2 Relations.- 2.3.3 Probability Potentials.- 2.3.4 Possibility Potentials.- 2.3.5 Spohn Potentials.- 2.3.6 Set Potentials.- 2.3.7 Gaussian Potentials.- 2.4 Partial Marginalization.- 3 Algebraic Theory.- 3.1 Congruences.- 3.2 Domain-Free Valuation Algebras.- 3.3 Subalgebras, Homomorphisms.- 3.3.1 Subalgebras.- 3.3.2 Homomorphisms and Isomorphisms.- 3.3.3 Weak Subalgebras and Homomorphisms.- 3.4 Null Valuations.- 3.5 Regular Valuation Algebras.- 3.6 Separative Valuation Algebras.- 3.7 Scaled Valuation Algebras.- 4 Local Computation.- 4.1 Fusion Algorithm.- 4.2 Collect Algorithm.- 4.3 Computing Multiple Marginals.- 4.4 Architectures with Division.- 4.5 Computations in Valuation Algebras with Partial Marginalization.- 4.6 Scaling and Updating.- 5 Conditional Independence.- 5.1 Factorization and Graphical Models.- 5.2 Conditionals in Regular Algebras.- 5.3 Conditionals in Separative Algebras.- 6 Information Algebras.- 6.1 Idempotency.- 6.2 Partial Order of Information.- 6.3 File Systems.- 6.4 Information Systems.- 6.5 Examples.- 6.5.1 Propositional Logic.- 6.5.2 Boolean Information Algebras.- 6.5.3 Linear Equations.- 6.5.4 Linear Inequalities.- 6.6 Compact Systems.- 6.7 Mappings.- 7 Uncertain Information.- 7.1 Algebra of Random Variables.- 7.1.1 Random Variables.- 7.1.2 Allocation of Probability.- 7.1.3 Support Functions.- 7.2 Probabilistic Argumentation Systems.- 7.2.1 Assumption-Based Information.- 7.2.2 Probabilistic Argumentation Systems.- 7.2.3 Operations with Argumentation Systems.- 7.3 Allocations of Probability.- 7.3.1 Algebra of Allocations.- 7.3.2 Normalized Allocations.- 7.3.3 Random Variables and Allocations.- 7.3.4 Labeled Allocations and their Algebra.- 7.4 Independent Sources.- 7.4.1 Independent Random Variables.- 7.4.2 Algebra of Bpas.- References.

On a class of uninorms

Abstract: Paper deals with binary operations in unit interval. We investigate connections between families of triangular norms, triangular conorms and uninorms. Certain structures of uninorms admits only the idempotent case.

Enright’s completions and injectively copresented modules

Abstract: Let 21 be a finite-dimensional simple Lie algebra over the complex numbers. It is shown that a module is complete (or relatively complete) in the sense of Enright if and only if it is injectively copresented by certain injective modules in the BGG-category O. Let A be the finite-dimensional algebra associated to a block of O. Then the corresponding block of the category of complete modules is equivalent to the category of eAe-modules for a suitable choice of the idempotent e. Using this equivalence, a very easy proof is given for Deodhar's theorem (also proved by Bouaziz) that completion functors satisfy braid relations. The algebra eAe is left properly and standardly stratified. It satisfies a double centralizer property similar to Soergel's combinatorial description of O. Its simple objects, their characters and their multiplicities in projective or standard objects are determined.

Idempotent semirings with a commutative additive reduct

Abstract: For every semigroup S , we define a congruence relation ρ on the power semiring (P(S),\cup,\circ) of S . If S is a band, then P(S)/ρ is an idempotent semiring . This enables us to find models for the free objects in the variety of idempotent semiring s whose additive reduct is a semilattice.

On the Riesz idempotent of class A operators

Abstract: In this paper, we show that if T is a class A operator and λ is a non–zero isolated eigenvalue of σ(T) , then EH = ker(T −λ) = ker(T −λ)∗ , where E is the Riesz idempotent with respect to λ . In this case, E is self–adjoint, i.e, it is an orthogonal projection. Mathematics subject classification (2000): 47A10, 47B20.

Normal semigroups of endomorphisms of proper independence algebras are idempotent generated

Abstract: LetAbe a proper independence algebra of finite rank, letGbe the group of automorphisms ofA ,l etabe a singular endomorphism and leta G be the semigroup generated by all the elementsg °1 ag ,w here g2G. The aim of this paper is to prove thata G is a semigroup

Combinatorial techniques for determining rank and idempotent rank of certain finite semigroups

Abstract: Let τ be a partition of the positive integer n. A partition of the set {1, 2 ,...,n } is said to be of type τ if the sizes of its classes form the partition τ of n. It is known that the semigroup S(τ ), generated by all the transformations with kernels of type τ , is idempotent generated. When τ has a unique non-singleton class of size d, the difficult Middle Levels Conjecture of combinatorics obstructs the application of known techniques for determining the rank and idempotent rank of S(τ ). We further develop existing techniques, associating with a subset U of the set of all idempotents of S(τ ) with kernels of type τ a directed graph D(U ); the directed graph D(U ) is strongly connected if and only if U is a generating set for S(τ ), a result which leads to a proof if the fact that the rank and the idempotent rank of S(τ ) are both equal to max n , n d +1 .

Idempotent (Asymptotic) Mathematics and the Representation Theory

Abstract: A brief introduction to Idempotent Mathematics is presented. Idempotent Mathematics can be treated as a result of a dequantization of the traditional Mathematics as the Planck constant tends to zero taking pure imaginary values. In the framework of Idempotent Mathematics some basic concepts and results of the theory of group representations (including some unexpected theorems of the Engel type) are discussed.

Multivalued linear projections

Abstract: A multivalued linear projection operator P defined on linear space X is a multivalued linear operator which is idempotent and has invariant domain. We show that a multivalued projection can be characterized in terms of a pair of subspaces and then establish that the class of multivalued linear projections is closed under taking adjoints and closures. We apply the characterizations of the adjoint and completion of projection together with the closed graph and closed range theorems to give criteria for the continuity of a projection defined ona normed linear space. A new proof of the theorem on closed sums of closed subspaces in a Banach space (cf. Mennicken and Sagraloof [9, 10]) follows as a simple corollary. We then show that the topological decomposition of a space may be expressed in terms of multivalued projections. The paper is concluded with an application to multivalued semi-Fredholm relations with generalized inverses. Mathematics Subject Classification (2000): 47A06, 47A53 Quaestiones Mathematicae 25 (2002), 503-512

Generalized compositions of fuzzy relations

Abstract: We examine compositions of fuzzy relations based on a binary operation *. We discuss the dependences between algebraic properties of the operation * and the induced sup - * composition. It is examined independently for monotone operations, for operations with idempotent, zero or identity element, for distributive and associative operations. Finally, we present consequences of these results for compositions based on triangular norms, triangular conorms and uninorms.

Nuclear semimodules and kernel theorems in idempotent analysis. An algebraic approach

Abstract: In this note we describe conditions under which, in idempotent functional analysis, linear operators have integral representations in terms of idempotent integral of V. P. Maslov. We define the notion of nuclear idempotent semimodule and describe idempotent analogs of the classical kernel theorems of L. Schwartz and A. Grothendieck. Our results provide a general description of a class of subsemimodules of the semimodule of all bounded functions with values in the Max-Plus algebra where some kind of kernel theorem holds, thus addressing an open problem posed by J. Gunawardena. Previously, some theorems on integral representations were obtained for a number of specific semimodules consisting of continuous or bounded functions taking values mostly in the Max-Plus algebra. In this work, a rather general case of semimodules over boundedly complete idempotent semirings is considered.

Soundness, idempotence and commutativity of set-sharing

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.

A new rank formula for idempotent matrices with applications

Abstract: A complex square matrix A is said to be idempotent, or a projector, whenever A2 = A; when A is idempotent, and Hermitian (or real symmetric), it is often called an orthogonal projector, otherwise an oblique projector. Projectors are closely linked to generalized inverses of matrices. For example, for a given matrix A the product PA = AA + is well known as the orthogonal projector on the range (column space) of A, where A+ is the Moore-Penrose inverse of A; which is the unique solution of the following four Penrose equations

An idempotent-analytic ISS small gain theorem with applications to complex process models

Abstract: vii

Congruences on *-Regular Semigroups

Abstract: In this paper we study the congruences of *-regular semigroups, involution semigroups in which every element is p-related to a projection (an idempotent fixed by the involution). The class of *-regular semigroups was introduced by Drazin in 1979, as the involutorial counterpart of regular semigroups. In the standard approach to *-regular semigroup congruences, one ,starts with idempotents, i.e. with traces and kernels in the underlying regular semigroup, builds congruences of that semigroup, and filters those congruences which preserve the involution. Our approach, however, is more evenhanded with respect to the fundamental operations of *-regular semigroups. We show that idempotents can be replaced by projections when one passes from regular to *-regular semigroup congruences. Following the trace-kernel balanced view of Pastijn and Petrich, we prove that an appropriate equivalence on the set of projections (the *-trace) and the set of all elements equivalent to projections (the *-kernel) fully suffice to reconstruct an (involution-preserving) congruence of a *-regular semigroup. Also, we obtain some conclusions about the lattice of congruences of a *-regular semigroup.

A classification of the maximal idempotent-generated subsemigroups of finite singular semigroups

Abstract: We describe the maximal idempotent-generated subsemigroups of the finite singular semigroup Singn on the finite set Xn={1,2, \ldots,n} , and count the number of its maximal idempotent-generated subsemigroups.

Unification modulo associativity and idempotency is NP-complete

Abstract: We show that the unification problem for the theory of one associative and idempotent function symbol (AI-unification), i.e. solving word equations in free idempotent semigroups, is NP-complete.

Semigroups with n + 1 Distinct Essentially n -Ary Term Operations

Abstract: In many papers idempotent algebras were characterized by their P n -sequences. This article as well as [19] yields the beginning of the investigation of nonidempotent algebras. Our aim is to give an equational description of all the semigroups without algebraic constants that have exactly n+1 different essentially n-ary term operations for every positive integer n. We prove that there are exactly two different varieties containing such the semigroups. The proof of this fact, as the reader can see, is not trivial and requires some work—mainly through lack of the idempotence of groupoids considered here.

Rational semimodules over the max-plus semiring and geometric approach of discrete event systems

Abstract: We introduce rational semimodules over semirings whose addition is idempotent, like the max-plus semiring, in order to extend the geometric approach of linear control to discrete event systems. We say that a subsemimodule of the free semimodule S^n over a semiring S is rational if it has a generating family that is a rational subset of S^n, S^n being thought of as a monoid under the entrywise product. We show that for various semirings of max-plus type whose elements are integers, rational semimodules are stable under the natural algebraic operations (union, product, direct and inverse image, intersection, projection, etc). We show that the reachable and observable spaces of max-plus linear dynamical systems are rational, and give various examples.

On bi-closed subbands of a band and free idempotent semirings

Abstract: The bi closed subbands of a band are introduced and its some properties and characterizations are given By using these results, the free object in the variety of idempotent semirings whose additive reducts are semilattices is established

Unification Modulo Associativity and Idempotency Is NP-complete

Abstract: We show that the unification problem for the theory of one associative and idempotent function symbol (AI-unification), i.e. solving word equations in free idempotent semigroups, is NP-complete.

Completeness of idempotent semifields and a generalization of the linear extension theorem in max-plus algebra

Abstract: Idempotent semifield is a fundamental algebraic structure in max-plus algebra. It differs from field in that the addition is idempotent and there is no inverse of addition. We discuss the completeness and complete extension of idempotent semifields, which is achieved through a max-plus version of Dedekind cuts. Subsequently, we use our results to obtain a generalization of the linear extension theorem, an important result in max-plus algebra and systems theory. This theorem had been proved valid for finitely generated semimodules, and we generalize it to countably generated semimodules.

Estimates for a mean time of trouble-free work for one queuing network class*

Abstract: The problem of estimation of the mean of process time for networks with queues and with synchronization of service requirements in nodes is considered Tl1e apparatus of the idempotent algebra is used which admits to describe the dynamics of system by the stochastic generalized difference equations For the cases that the network topology is described by an acyclic graph, the upper and lower estimates of mean time of work are obtained In the present work the problem of estimating a mean time of the trouble-free work for queuing networks with the synchronization of requirements in nodes is considered The apparatus of idempotent algebra is used which permits us to describe the system’s dynamics by the generalized stochastic difference equations For the case that the net topology is described by an acyclic graph, the upper and lower estimates of a mean time of trouble-free work are obtained 1 Idempotent algebra Consider the set of real numbers E, extended by adding one element 5 = -oo, with the defined in it operations 69 and ®, whose definitions for any sv, y G E are the following: