Total order

... group communication, fault-tolerance, agreement problems, message passing, total ordering, global ordering, atomic multicast, atomic broadcast, classification, taxonomy ... Part of this research was conducted for the program “Fostering Talent in Emergent Research Fields ...

We begin by constructing the algebra of classical intervals and prove that it is a nondistributive abelian semiring. Next, we formalize the notion of interval dependency, along with discussing the algebras of two alternate theories of intervals: modal intervals, and constraint intervals. With a view to treating some problems of the present interval theories, we present an alternate theory of intervals, namely the "theory of optimizational intervals", and prove that it constitutes a rich S-field algebra, which extends the ordinary field of the reals, then we construct an optimizational complex interval algebra. Furthermore, we define an order on the set of interval numbers, then we present the proofs that it is a total order, compatible with the interval operations, dense, and weakly Archimedean. Finally, we prove that this order extends the usual order on the reals, Moore's partial order, and Kulisch's partial order on interval numbers.

Keywords. Classical interval arithmetic; Machine interval arithmetic; Interval dependency; Constraint intervals; Modal intervals; Classical complex intervals; Optimizational intervals; Optimizational complex intervals; S-field algebra; Ordering interval numbers.

In this article we consider the Non-linear single varible Fuzzy-valued functions with regards to the total order relation defined on Fuzzy number space on the lines of [9]. Using this order relation and Hukuhara differentiability of fuzzy-valued functions we obtain the necessary and sufficient conditions for optimality of nonlinear single variable fuzzy-valued funcions.

The performance of a distributed system is affected by the various functions of its components. The interaction between components such as network nodes, computer systems and system programs is examined with special interest accorded to its effect on system reliability. At affordable time and space costs, the analytic hierarchy process (AHP) is used to determine how the reliability of a distributed system may be controlled by appropriately assigning weights to its components. Illustrative case studies, that display the system structure, the assignment of weights and the AHP handling are presented.

Causally and totally ordered communication support (CATOCS) has been proposed as important to provide as part of the basic building blocks for constructing reliable distributed systems. In this paper, we identify four major limitations to CATOCS, investigate the applicability of CATOCS to several classes of distributed applications in light of these limitations, and the potential impact of these facilities on communication scalability and robustness. From this investigation, we find limited merit and several potential problems in using CATOCS. The fundamental difficulty with the CATOCS is that it attempts to solve state problems at the communication level in violation of the

A general algebraic method for decoding all types of binary cyclic codes is presented. It is shown that such a method can correct t=[(d-1)/2] errors, where d is the true minimum distance of the given cyclic code. The key idea behind this decoding technique is a systematic application of the algorithmic procedures of Grobner bases to obtain the error-locator polynomial L(z). The discussion begins from a set of syndrome polynomials F and the ideal T(F) generated by F. It is proved here that the process of transforming F to the normalized reduced Grobner basis of I(F) with respect to the “purely lexicographical” ordering automatically converges to L(z). Furthermore, it is shown that L(z) can be derived from any normalized Grobner basis of I(F) with respect to any admissible total ordering. To illustrate this new approach, the procedures for decoding certain BCH codes and quadratic residue codes are demonstrated

We extend the concept of Arens regularity of a Banach algebra \(\mathcal{A}\) to the case that there is an \(\mathcal{O}\) -module structure on \(\mathcal{A}\) , and show that when S is an inverse semigroup with totally ordered subsemigroup E of idempotents, then A=ℓ 1(S) is module Arens regular if and only if an appropriate group homomorphic image of S is finite. When S is a discrete group, this is just Young’s theorem which asserts that ℓ 1(S) is Arens regular if and only if S is finite.

We investigate the ideal structure of the Toeplitz algebra $ \mathcal{T}(\Gamma) $ of a totally ordered abelian group $ \Gamma $ . We show that the primitive ideals of $ \mathcal{T}(\Gamma) $ are parametrised by the disjoint union $ X $ of the duals $ \hat{I} $ of the order ideals $ I $ of $ \Gamma $ , and identify the hull-kernel topology on $ X $ when the chain of orderideals in $ \Gamma $ is isomorphic to a subset of $ \{-\infty\}\cup \mathbb{Z} \cup \{\infty\} $

Abstract\ Ve cOllf: iider multi-criteria f: iequent, ial decision making problems where the vcctor-" valucd evaluations arc compared by a given, fixed total ordering. Condit. ions for the opt. irnalit y of statiOIl<-l, r}'p () lich s;-weI the Bellman opti malit, y equation are given for a. special, but. important class of problems''v hell the eval uation of policies can be computed for the criteria, independently of each other. The anal)'sis requires special cafC as t. he t. opol ag, Y int. roduced by polnL\ visc convergence a. ncl the or< 1cr-topology ...

We provide a pure algebraic version of the dynamical characterization of Conrad's property. This approach allows dealing with general group actions on totally ordered spaces. As an application, we give a new and somehow constructive proof of a theorem first established by Linnell: an orderable group having infinitely many orderings has uncountably many. This proof is achieved by extending to uncountable orderable groups a result about orderings which may be approximated by their conjugates. This last result is illustrated by an example of an exotic ordering on the free group given in the Appendix.