Karim Belaïd, Bahri Cherif, Othman Echi · 2023
I. Kaplansky [5] states the following problem " Under what conditions a partially ordered set is isomorphic to the prime spectrum of a ring ordered by inclusion ?". W.J. Lewis and J. Ohm called a partially ordered set spectral if it is order isomorphic to Spec(R) for some ring R. Although there are partial results about this subject [2], [3], [6] and [7], the problem of characterizing the spectral sets remains open. However the corresponding topological problem was completely answered by M. Hochster in his remarkable paper [4]. Let (X, ≤) be a partially ordered set, a topology τ on X is said to be compatible with the ordering ≤ if the closure of {x} is cl{ x } = [ x, → [ = { y ∈ X / x ≤ y }, which is equivalent to the following two conditions: [x, → [is a closed subset of (X, τ), for all x ∈ X. every closed subset F of (X, τ) is closed under specialization (i.e: [ x, → [ ⊆ F, for all x ∈ F).
Cited by 1
Studies in logic and the foundations of mathematics · 2002
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Lecture notes in computer science · 1979
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William Zhu, Fei‐Yue Wang · Lecture notes in computer science · 2006
Cited by 46
Hongwei Zhao, Zhen Wang, Pingping Liu · Pattern Recognition · 2015
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Lemnaouar Zedam, Raúl Pérez‐Fernández, Hassane Bouremel et al. · International Journal of General Systems · 2018
In a recent paper, we have introduced the notion of clone relation of a given binary relation. Intuitively, two elements are said to be “clones” if they are related in the same way w.r.t. every other element. In this paper, we generalize this notion from pairs of elements to sets of elements of any cardinality, resulting in the introduction of clonal sets. We investigate the most important properties of clonal sets, paying particular attention to the introduction of the clonal closure operator, to the analysis of the (lattice) structure of the set of clonal sets and to a distance metric expressing how close two elements are to being clones.
Cited by 2
Jean‐Philippe Chancelier, Michel De Lara, Benjamin Heymann · arXiv (Cornell University) · 2021
The concept of d-separation holds a pivotal role in causality theory, serving\nas a fundamental tool for deriving conditional independence properties from\ncausal graphs. Pearl defined the d-separation of two subsets conditionally on a\nthird one. In this study, we present a novel perspective by showing i) how the\nd-separation can be extended beyond acyclic graphs, possibly infinite, and ii)\nhow it can be expressed and characterized as a binary relation between\nvertices. Compared to the typical perspectives in causality theory, our\nequivalence opens the door to more compact and computational proofing\ntechniques, because the language of binary relations is well adapted to\nequational reasoning. Additionally, and of independent interest, the proofs of\nthe results presented in this paper are checked with the Coq proof assistant.\n
Cited by 2Open access
Sik-Sang Yau, P. C. Grabow, Bob P. Weems · 1982
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Liping An · 2011
Classical definitions of lower and upper approximations,which are core concepts of rough sets theory,are based on indiscernibility relations.This relation is well suited in the case of nominal attributes.However,many real-world problems often involve both qualitative and quantitative attributes. It is necessary to generalize indiscernibility relation by using some other binary relations.The binary relations defined from some similarity measures at the level of any single attribute are considered.These relations are aggregated into the global binary relations at the level of the set of attributes.Then,the rough approximations and the boundaries of the unions of decision classes are defined using the global binary relations.And,the operations of approximations and boundaries are defined to describe the certain, possible and doubtful knowledge.The operations satisfy the properties of rough inclusion,complementarity, identity of boundaries,and monotonicity.The new rough set approach can detect the inconsistency which may occur when in the decision table there are both qualitative and quantitative attributes.
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Robert L. Blair · Duke Mathematical Journal · 1955
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Yang Liu, Dietmar Block · Computer Physics Communications · 2024
The applicability of the Stokes-Einstein (SE) relation in two-dimensional finite binary mixtures is tested using the Langevin simulation method. Calculations of viscosity and self-diffusion coefficients are compared between monodisperse and binary systems. It is shown that adapted definitions of coupling strength Γ and screening parameter κ allow to describe the transport properties for both monodisperse and binary systems consistently. Further, it is discussed how the mixing ratio and charge ratio of the binary mixture affects transport properties.
Cited by 5Open access
Grekovs, Romans · OpenGrey (Institut de l'Information Scientifique et Technique) · 2003
This work represents the result of theoretical and experimental research, developing and modifications of the methods for the tasks of fuzzy pattern recognitions. The great value in recognition tasks has the selection of such set of features, which is able to increase maximally recognition ability of the system. The present work represents qualitatively new solution for the task of feature efficiency evaluation in the fuzzy pattern recognition, which uses such objects of the space as direct and supplementary shadows. The work's basic results are described in the present theses, which consist of the six chapters: Chapter 1 gives a short description of the basic directions of the research. Chapter 2 is dedicated to the developing of an algorithm of classification on the basis of effective feature set. Its steps and the ways of developing are clarified. Chapter 3 is dedicated to the reproducing and evaluations of compositions of binary relations using graphs. Also some information of the graph theory, which gives the direct representation of the graph ideas and the operations over them is given. The binary composition operation's representation in the view of the graphs is described as well. Chapter 4 describes the supplementary shadows of fuzzy sets and their applications to the task of pattern recognition. Also the possibility of the usage of supplementary shadows in the tasks of fuzzy pattern recognition for improving the quality of recognition is represented. Chapter 5 offers the variant of the modification of the method of recognition with participation of the compositions of binary relations to the n-dimensional description spaces of features. Chapter 6 presents an interpretation of the method of Cross Validation for the task of classification of fuzzy objects on the basis of effective feature set. Conclusion contains the characteristics of the work. Main achievements of the work and some disadvantages of the proposed methods are summarised. Also the future directions of the work are described. As the result of the practical work the software described in Appendix is developed using the Microsoft Development Studio 6.0. The main results of work were authorised and published in the international journals and conferences.
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Xu Li, Jianghua Wang, Chunling Wu et al. · 2021
The study of attribute reduction in relation decision systems is an important issue in the field of artificial intelligence. A relation decision system can be constructed when the binary relation is not required to meet reflexivity, symmetry and transitivity. Attribute reduction is to delete irrelevant or redundant attributes according to specific rules. In this paper, according to the existing concept of relative indiscernibility relation set under the equivalence relation, the concept of a binary relation set base is proposed in the relation decision system. Secondly, the concept of the corresponding binary relation set base reduction is proposed, and then the corresponding discernibility matrix that is proved is proposed. Thus, a binary relation base reduction algorithm of the corresponding discernibility matrix is proposed. An example is given to illustrate the effectiveness of the algorithm proposed in this paper.
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Krokhin, A., Schweigert, D. · Durham Research Online (Durham University) · 2001
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Malkov, O. Yu., Малков, О. Ю. · Electronic scientific archive of UrFU (Ural Federal University) · 2019
Observational types of binary stars for which components dynamical masses could be determined, are discussed. These data serve as a unique source for construction of the mass-luminosity relation, a fundamental law, which is especially important for getting of the initial mass function, which in turn fully determines structure and evolution of stellar systems.
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Osama Sayed, Nabil Hassan Sayed · SHILAP Revista de lepidopterología · 2022
In the study of partially ordered sets, topologies such as Scott-topology have shown to be of paramount importance. In order to have analogous topology-like tools in the more general setting of quantitative domains, we introduce a method to construct Scotttopology on a set equipped with a transitive binary relation which we call t-set. As an application of this result there is a Scott-topology associated to any topology induced by its specialization pre-ordered relation. Some relations between this topology and the original topology are investigated.
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飯島 淳一, Iijima Junichi, 柴 直樹 et al. · Tokyo Tech Research Repository (Tokyo Institute of Technology) · 2006
In systems theory, we often use the concept of dependence. There are several research to construct a unified framework for dependence. Mainly, the concept of dependence is defined with a dais of subsets, where the concept of algebraic closure systems is often used. \n\nIn this paper, we define dependence as a binary relation, which we call dependence structure, and develop a similar argument as for the dependence with a class of subsets. And we will clarify a relationship between dependence structure and the usual concept of dependence. \n\nWe also show what conditions assure the existence of a basis for a dependence structure. \n\nThe concept of functional dependence in database theory can be represented as a dependence structure and we will show that a key for a relational scheme is a basis for the dependence structure.
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Jérémy Barbay, Francisco Claude, Gonzalo Navarro · arXiv (Cornell University) · 2012
Binary relations are an important abstraction arising in many data representation problems. The data structures proposed so far to represent them support just a few basic operations required to fit one particular application. We identify many of those operations arising in applications and generalize them into a wide set of desirable queries for a binary relation representation. We also identify reductions among those operations. We then introduce several novel binary relation representations, some simple and some quite sophisticated, that not only are space-efficient but also efficiently support a large subset of the desired queries.
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Юрий Казаков · SSRN Electronic Journal · 2025
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Jérémy Barbay, Francisco Claude, Gonzalo Navarro · Information and Computation · 2013
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