Partial Order Relation Research Articles

On soft topological ordered spaces

In this paper, the authors initiate a soft topological ordered space by adding a partial order relation to the structure of a soft topological space. Some concepts such as monotone soft sets and increasing (decreasing) soft operators are presented and their main properties are studied in detail. The notions of ordered soft separation axioms, namely p-soft Ti-ordered spaces (i=0,1,2,3,4) are introduced and the relationships among them are illustrated with the help of examples. In particular, the equivalent conditions for p-soft regularly ordered spaces and soft normally ordered spaces are given. Moreover, we define the soft topological ordered properties and then verify that the property of being p-soft Ti-ordered spaces is a soft topological ordered property, for i=0,1,2,3,4. Finally, we investigate the relationships between soft compactness and some ordered soft separation axioms and point out that the condition of soft compactness is sufficient for the equivalent between p-soft T2-ordered spaces and p-soft T3-ordered spaces.

Toward an algebraic theory of systems

We propose the concept of a system algebra with a parallel composition operation and an interface connection operation, and formalize composition-order invariance, which postulates that the order of composing and connecting systems is irrelevant, a generalized form of associativity. Composition-order invariance explicitly captures a common property that is implicit in any context where one can draw a figure (hiding the drawing order) of several connected systems, which appears in many scientific contexts. This abstract algebra captures settings where one is interested in the behavior of a composed system in an environment and wants to abstract away anything internal not relevant for the behavior. This may include physical systems, electronic circuits, or interacting distributed systems.One specific such setting, of special interest in computer science, are functional system algebras, which capture, in the most general sense, any type of system that takes inputs and produces outputs depending on the inputs, and where the output of a system can be the input to another system. The behavior of such a system is uniquely determined by the function mapping inputs to outputs. We consider several instantiations of this very general concept. In particular, we show that Kahn networks form a functional system algebra and prove their composition-order invariance.Moreover, we define a functional system algebra of causal systems, characterized by the property that inputs can only influence future outputs, where an abstract partial order relation captures the notion of “later”. This system algebra is also shown to be composition-order invariant and appropriate instantiations thereof allow to model and analyze systems that depend on time.

Heyting Almost Distributive fuzzy Lattices

In this paper, we introduce the concept of Heyting almost distributive fuzzy lattices (HADFL) using the concepts of Heyting almost distributive lattices (HADL), almost distributive fuzzy lattices, fuzzy partial order relation and fuzzy Heyting algebra. Using the properties of fuzzy Heyting algebra, we also give a necessary and sufficient condition for an HADFL to be fuzzy Heyting algebra (FHA).

Concept coupling learning for improving concept lattice-based document retrieval

The semantic information in any document collection is critical for query understanding in information retrieval. Existing concept lattice-based retrieval systems mainly rely on the partial order relation of formal concepts to index documents. However, the methods used by these systems often ignore the explicit semantic information between the formal concepts extracted from the collection. In this paper, a concept coupling relationship analysis model is proposed to learn and aggregate the intra- and inter-concept coupling relationships. The intra-concept coupling relationship employs the common terms of formal concepts to describe the explicit semantics of formal concepts. The inter-concept coupling relationship adopts the partial order relation of formal concepts to capture the implicit dependency of formal concepts. Based on the concept coupling relationship analysis model, we propose a concept lattice-based retrieval framework. This framework represents user queries and documents in a concept space based on fuzzy formal concept analysis, utilizes a concept lattice as a semantic index to organize documents, and ranks documents with respect to the learned concept coupling relationships. Experiments are performed on the text collections acquired from the SMART information retrieval system. Compared with classic concept lattice-based retrieval methods, our proposed method achieves at least 9%, 8% and 15% improvement in terms of average MAP, IAP@11 and P@10 respectively on all the collections.

Low Complexity Method for Channel-Dependent Construction of Polar Codes Using Partial Order

In this paper, a novel low-complexity method for channel-dependent construction of polar codes is designed by exploiting the partial order (PO) relation to eliminate redundant computations. The proposed method selects the information set with a threshold, thus avoiding the reliability computation of unnecessary synthetic channels and the cost of sorting. We first introduce two key iterative strategies based on the PO relation of the synthetic channels. Then, we analyze the iteration process of the proposed method through a binary tree and a linear table structure and further derive the theoretical complexity of the proposed method. We validate the theoretical analysis through simulation results. Finally, considering the practical application, we propose a suboptimal simplified version of the proposed method to significantly reduce the complexity. Compared with the sorting method, the proposed simplified method needs only at most 10% of the comparison numbers in the process of the information set selection with negligible performance loss under the cyclic redundancy check-aided successive cancellation list decoding.

A Study on Some Fundamental Properties of Continuity and Differentiability of Functions of Soft Real Numbers

We introduce a new type of functions from a soft set to a soft set and study their properties under soft real number setting. Firstly, we investigate some properties of soft real sets. Considering the partial order relation of soft real numbers, we introduce concept of soft intervals. Boundedness of soft real sets is defined, and the celebrated theorems like nested intervals theorem and Bolzano-Weierstrass theorem are extended in this setting. Next, we introduce the concepts of limit, continuity, and differentiability of functions of soft sets. It has been possible for us to study some fundamental results of continuity of functions of soft sets such as Bolzano’s theorem, intermediate value property, and fixed point theorem. Because the soft real numbers are not linearly ordered, several twists in the arguments are required for proving those results. In the context of differentiability of functions, some basic theorems like Rolle’s theorem and Lagrange’s mean value theorem are also extended in soft setting.

On Analysis of Tangshan Pre-Hospital Service Efficiency Based on Improved Fuzzy-DEA

An allocation model based on improved Fuzzy-DEA is proposed for Tangshan medical health resource. The membership function is particularly introduced to characterize the non-strict partial order relation of the fuzzy numbers, and the internal form of the unit efficiency values is converted into point values. The variance analysis index system is then used to find factors leading to non-DEA effectiveness and the allocation plan is recommended accordingly: the relevant departments should coordinate the inter-regional resources and optimize the medical staffing level.

Partial order relations on family of sets and scalarizations for set optimization

In this study, some new order relations on family of sets are introduced by using Minkowski difference. The relations between these orders and the ordering cone of the vector space are obtained. It is shown that depending on the corresponding cone, these order relations are partial orders on the family of nonempty bounded sets. Some relationships between these order relations and upper and lower set less order relations are investigated. Also, two scalarizing functions are introduced in order to replace set optimization problems with respect to these partial order relations with scalar optimization problems. Moreover, necessary and sufficient optimality conditions are presented.

Resolve conflicting rankings of outcomes in network meta-analysis: Partial ordering of treatments.

Network meta-analysis has evolved into a core method for evidence synthesis in health care. In network meta-analysis, 3 or more treatments for a given medical condition are compared, based on a number of clinical studies, usually randomized controlled trials. Often, many different endpoints are investigated, related to different aspects of the patient's outcome, such as efficacy, safety, acceptability, or costs of a treatment. Different outcomes may lead to different rankings of the treatments. We use the existing theory of partially ordered sets and show how the relations between the treatments in a network meta-analysis can be illustrated by Hasse diagrams, that is, directed graphs showing the partial order relations, and by structured scatter plots and biplots.

Kinds of [formula omitted]-fuzzy Filters of Fuzzy Lattices

The starting point of this work comes from the results obtained by Inheung Chon in his paper “Fuzzy partial order relations and fuzzy lattices”. In this way, some characterizations of filters and prime filters are given. Also, we characterize fuzzy t-filters and fuzzy prime t-filters using their α-cuts. Finally, we give necessary and sufficient conditions under which a fuzzy set is a prime t-filter, presenting the notion of fuzzy lattices isomorphism.

Language Equations for Maximal Decompositions in Coordination Control

The problem considered is to construct all solutions of an equation for a tuple of languages. The tuple in synchronous composition should equal a considered language. Of special interest are the maximal solutions with respect to a partial order relation on the set of solutions. The motivation of the problem is coordination control. The approach to the problem is to transform the equation to a fixed-point equation. An algorithm is proposed which constructs tuples of maximal solutions by convergence of a sequence of tuples to a solution of the fixed-point equation. Modular supervisory control is shown to benefit from the proposed approach.

Notes on covering-based rough sets from topological point of view: Relationships with general framework of dual approximation operators

In the recent article “On some types of covering rough sets from topological points of view” [14], the author develops a topological approach to covering-based rough sets. In this context, a number of corresponding approximation operators are introduced, their inclusion relationships are verified, and various conditions under which the operators coincide are proven.On the other hand, a lot of effort has recently been dedicated by several authors to study covering-based approximation operators within a general framework of dual approximation operators [2,3,7–10,12].In this note, we study correspondences between the framework of Zhao and the framework established in [2]. In particular, we evaluate how the newly introduced topological approximation operators relate to existing ones in terms of equalities and partial order relations.

Computing the probability of union in the [formula omitted]-dimensional Euclidean space for application of the multivariate quantile: [formula omitted]-level efficient points

As recently studied in Lee and Prékopa (2017), in the general case of Rn,n≥3, if certain conditions among the projected sets are met, the probability of union of such sets can be efficiently calculated by the modified inclusion–exclusion formula in a recursive manner. In this paper, we study with the fullest generality how to construct such suitable conditions on the partial order relation. Upon this basis we can use the modified inclusion–exclusion formulas in any case. Numerical examples are presented.

Order algebras: a quantitative model of interaction

A quantitative model of concurrent interaction is introduced. The basic objects are linear combinations of partial order relations, acted upon by a group of permutations that represents potential non-determinism in synchronisation. This algebraic structure is shown to provide faithful interpretations of finitary process algebras, for an extension of the standard notion of testing semantics, leading to a model that is both denotational (in the sense that the internal workings of processes are ignored) and non-interleaving. Constructions on algebras and their subspaces enjoy a good structure that make them (nearly) a model of differential linear logic, showing that the underlying approach to the representation of non-determinism as linear combinations is the same.

On relationships among different types of solutions of fuzzy optimization problems

Since fuzzy optimization problems are extensively studied by considering the endpoint functions of level sets of fuzzy-valued objective function with respect to the fuzzy-max order, in this paper, by defining two related partial order relations we establish the complete relationships between the op timal solutions with respect to the fuzzy-max order and the ones with respect to the related partial order relations. These relationships reflect the intrinsic properties of fuzzy optimizations.

Interval-valued upside potential and downside risk portfolio optimisation

A novel interval optimisation approach is developed to include imprecise forecasts into the portfolio selection process for investors measuring upside potential and downside risk as deviations from a target return. Crisp scenarios are substituted by interval scenarios and the resulting interval optimisation problem is solved in a tractable manner by means of a bi-objective formulation exploiting a partial order relation between intervals. Four utility case studies involving assets from the F.T.S.E. M.I.B. Index are considered to illustrate how impreciseness can be efficiently handled in portfolio management.

Fat node leading tree for data stream clustering with density peaks

Detecting clusters of arbitrary shape and constantly delivering the results for newly arrived items are two critical challenges in the study of data stream clustering. However, the existing clustering methods could not deal with these two problems simultaneously. In this paper, we employ the density peaks based clustering (DPClust) algorithm to construct a leading tree (LT) and further transform it into a fat node leading tree (FNLT) in a granular computing way. FNLT is a novel interpretable synopsis of the current state of data stream for clustering. New incoming data is blended into the evolving FNLT structure quickly, and thus the clustering result of the incoming data can be delivered on the fly. During the interval between the delivery of the clustering results and the arrival of new data, the FNLT with blended data is granulated as a new FNLT with a constant number of fat nodes. The FNLT of the current data stream is maintained in a real-time fashion by the Blending-Granulating-Fading mechanism. At the same time, the change points are detected using the partial order relation between each pair of the cluster centers and the martingale theory. Compared to several state-of-the-art clustering methods, the presented model shows promising accuracy and efficiency.

A Computer Approach to Mathematics Curriculum Developments Debugging

Background:Sequencing contents is of great importance for instructional design within the teaching planning processes. We believe that it is key for a meaningful learning.Material and methods:Therefore, we propose to formally establish a partial order relation among the contents: the relation: “to be a prerequisite”. We have applied this approach to the mathematical contents of the compulsory Secondary Education of the Spanish educational system. This information has been modeled as a graph. The amount of contents considered (814) and the number of ordered pairs in the order relation considered (17,782) has produced a big graph. In order to work effectively with that amount of data, we have used software specialized in network analysis. More precisely, we have used the software packages Pajek and Gephi.Results:This software, together with the use of techniques borrowed from graph theory, has allowed providing a tool for debugging curriculum developments (similar to rule based expert systems verification).Conclusions:This is a contribution to the processes of sequencing contents and, therefore, to the processes of planning teaching and instructional design.

DRank: A semi-automated requirements prioritization method based on preferences and dependencies

Abstract There are many types of dependencies between software requirements, such as the contributions dependencies (Make, Some+, Help, Break, Some-, Hurt) and business dependencies modeled in the i * framework. However, current approaches for prioritizing requirements seldom take these dependencies into consideration, because it is difficult for stakeholders to prioritize requirements considering their preferences as well as the dependencies between requirements. To make requirement prioritization more practical, a method called DRank is proposed. DRank has the following advantages: 1) a prioritization evaluation attributes tree is constructed to make the ranking criteria selection easier and more operable; 2) RankBoost is employed to calculate the subjective requirements prioritization according to stakeholder preferences, which reduces the difficulty of evaluating the prioritization; 3) an algorithm based on the weighted PageRank is proposed to analyze the dependencies between requirements, allowing the objective dependencies to be automatically transformed into partial order relations; and 4) an integrated requirements prioritization method is developed to amend the stakeholders’ subjective preferences with the objective requirements dependencies and make the process of prioritization more reasonable and applicable. A controlled experiment performed to validate the effectiveness of DRank based on comparisons with Case Based Ranking, Analytical Hierarchy Process, and EVOLVE. The results demonstrate that DRank is less time-consuming and more effective than alternative approaches. A simulation experiment demonstrates that taking requirement dependencies into consideration can improve the accuracy of the final prioritization sequence.

Convex congruences

For an algebra {mathbf {A}} belonging to a quasivariety {mathcal {K}}, the quotient {mathbf {A}}/Theta need not belong to {mathcal {K}} for every Theta in {{mathrm{Con}}}~{mathbf {A}}. The natural question arises for which Theta in {{mathrm{Con}}}~{mathbf {A}}, {mathbf {A}}/Theta in {mathcal {K}}. We consider algebras {mathbf {A}}=(A,rightarrow ,1) of type (2, 0) where a partial order relation is determined by the operations rightarrow and 1. Within these, we characterize congruences on {mathbf {A}} for which {mathbf {A}}/Theta belongs to the same quasivariety as {mathbf {A}}. In several particular cases, these congruences are determined by the property that every class is a convex subset of A.