On operations of soft sets

Soft set theory, proposed by Molodtsov, has been regarded as an effective mathematical tool to deal with uncertainties. In this paper, first we prove that certain De Morgan’s law hold in soft set theory with respect to different operations on soft sets. Then, we discuss the basic properties of operations on soft sets such as intersection, extended intersection, restricted union and restricted difference. Moreover, we illustrate their interconnections between each other. Also we define the notion of restricted symmetric difference of soft sets and investigate its properties. The main purpose of this paper is to extend the theoretical aspect of operations on soft sets.     

软Vague关系再研究

给出了软Vague关系的广义交、狭义交、狭义并、相对补运算的概念,并分别研究了这些运算的若干代数性质。     

Vague软集的一些代数性质

基于Vague集和软集现有理论以及Vague集思想与软集思想之间的联系,初步提出了Vague软集的狭义交、狭义并、相对补运算的概念,并在此基础上分别研究了这些运算的若干代数性质。     

Possibility neutrosophic soft sets and PNS-decision making method

In this paper, we introduce concept of possibility neutrosophic soft set and define some related concepts such as possibility neutrosophic soft subset, possibility neutrosophic soft null set, and possibility neutrosophic soft universal set. Then, based on definitions of n-norm and n-conorm, we define set theoretical operations of possibility neutrosophic soft sets such as union, intersection and complement, and investigate some properties of these operations. We also introduce AND-product and OR-product operations between two possibility neutrosophic soft sets. We propose a decision making method called possibility neutrosophic soft decision making method (PNS-decision making method) which can be applied to the decision making problems involving uncertainty based on AND-product operation. We finally give a numerical example to display application of the method that can be successfully applied to the problems.     

新型软亚BCI-代数的进一步研究

提出了两个亚BCI-代数的并代数、两个软集在并代数上的扩展交、软集的软平移以及两个软集的合成运算等新概念。举例说明了并代数以及在并代数上扩展交的存在性。证明了两个新型软亚BCI-代数在一定的条件下在并代数上的扩展交仍然是新型软亚BCI-代数。研究了新型软亚BCI-代数的软平移及投影等的相关性质。另外,利用两个软集的合成运算、软集的水平集以及广义特征函数分别给出了新型软亚BCI-代数的等价刻画。     

Compatibility of t-norms with the concept of ?-partition

In this paper we study the behaviour of a kind of partitions formed by fuzzy sets, the ?-partitions, with respect to three important operations: refinement, union and product of partitions. In the crisp set theory, the previous operations lead to new partitions: every refinement of a partition is also a partition; the union of partitions of disjoint sets is a partition of the union set; the product of two partitions of two sets is a partition of the intersection of the partitioned sets. It has been proven that ?-partitions extend the three previous properties when the intersection of fuzzy sets is defined by the minimum t-norm and the union by the maximum t-conorm. In this paper we consider any t-norm defining the intersection of fuzzy sets and we characterize those t-norms for which refinements, unions and products of ?-partitions are ?-partitions. We pay special attention to these characterizations in the case of continuous t-norms.     

双极值模糊软集

提出了双极值模糊软集的概念,给出双极值模糊软集的补、并、交、"且"及"或"运算,并讨论了它们的性质。     

Soft set theory

In this paper, the authors study the theory of soft sets initiated by Molodtsov. The authors define equality of two soft sets, subset and super set of a soft set, complement of a soft set, null soft set, and absolute soft set with examples. Soft binary operations like AND, OR and also the operations of union, intersection are defined. De Morgan's laws and a number of results are verified in soft set theory.     

An evidential reasoning approach to attribute value conflictresolution in database integration

Resolving domain incompatibility among independently developed databases often involves uncertain information. DeMichiel (1989) showed that uncertain information can be generated by the mapping of conflicting attributes to a common domain, based on some domain knowledge. We show that uncertain information can also arise when the database integration process requires information not directly represented in the component databases, but can be obtained through some summary of data. We therefore propose an extended relational model based on Dempster-Shafer theory of evidence to incorporate such uncertain knowledge about the source databases. The extended relation uses evidence sets to represent uncertainty in information, which allow probabilities to be attached to subsets of possible domain values. We also develop a full set of extended relational operations over the extended relations. In particular, an extended union operation has been formalized to combine two extended relations using Dempster's rule of combination. The closure and boundedness properties of our proposed extended operations are formulated. We also illustrate the use of extended operations by some query examples     

软BCK代数

提出了软BCK代数的概念,研究了软BCK代数的并,交和乘积运算,讨论了这些运算的代数性质。     

区间复模糊软集的距离测度及其应用

基于区间复模糊软集的概念,定义了多种区间复模糊软集的距离测度公式,包含Hausdorff距离、Hamming距离、Euclidean距离、广义Hausdorff距离、广义Euclidean距离、广义加权Hausdorff距离、广义加权Euclidean距离、加权Hausdorff距离、加权Hamming距离、加权Euclidean距离。提出了除交、并、补运算外的区间复模糊软集的加法、乘法、部分隶属度和部分非隶属度运算以及距离测度之间的运算性质。基于区间复模糊软集距离测度构造了一种TOPSIS决策方法,并将这种决策方法应用于经济分析中,验证了所提方法的可行性。     

软Vague关系

提出软Vague关系的一些基本概念,研究了软Vague关系的∨、∧以及并、交、补代数运算的若干性质。     

Pawlak近似空间中软集的软上（下）近似

在Pawlak近似空间上首次引入了等效软集的软上下近似的概念,研究了软上下近似的交、并、补代数运算的若干性质。结果令人满意。     

Operations on fuzzy sets—an axiomatic approach

We introduce intersection and union of fuzzy sets, constructed by means of operations on the unit interval which may depend on the considered element of the universe of discourse. A similar approach for complementation was given by Lowen [7]. The main result is that element-dependent operations are not compatible with the preservation of the intersection and union under the inverse image of a function. Finally we consider fuzzy σ-algebras built with these set-theoretic operations.     

On fuzzy complements

Bellman and Giertz [1] showed that under reasonable restrictions the generalized operations of union and intersection for fuzzy sets have to be l.u.b. and g.l.b. in I^x endowed with the usual order, where I denotes the unit interval and X is an arbitrary set. They also remarked that an analogous result for fuzzy complementation was not readily obtainable. It is the purpose of this paper to characterize those operations which are acceptable as complementation. We define a category of fuzzy complemented spaces and see which subcategories have properties analogous to those of the category of sets and functions. We also consider some topological aspects of the matter.     

A new complicated-knowledge representation approach based on knowledge meshes

This paper presents a new complicated-knowledge representation method for the self-reconfiguration of complex systems such as complex software systems, complex manufacturing systems, and knowledgeable manufacturing systems. Herein, new concepts of a knowledge mesh (KM) and an agent mesh (AM) are proposed along with a new KM-based approach to complicated-knowledge representation. KM is the representation of such complicated macroknowledge as an advanced manufacturing mode, focusing on knowledge about the structure, functions, and information flows of an advanced manufacturing system. The multiple set, KM, and the mapping relationships between both, are then formally defined. The union, intersection, and minus operations on the multiple sets are proposed, and their properties proved. Then, the perfectness of a KM, the redundancy set between the two KMs, and the multiple redundancy set on the redundancy set are defined. Three examples are provided to illustrate the concepts of the KM, multiple set, multiple redundancy set, and logical operations. On the basis of the above, the KM-based inference engine is presented. In logical operations on KMs, each KM is taken as an operand. A new KM obtained by operations on KM multiple sets can be mapped into an AM for automatic reconfiguration of complex software systems. Finally, the combination of two real management modes is exemplified for the effective application of the new KM-based method to the self-reconfiguration of complex systems. It is worth mentioning that KM multiple sets can also be taken as a new formal representation of software systems if their corresponding AMs are the real software systems.     

A novel soft rough set: Soft rough hemirings and corresponding multicriteria group decision making

In this paper, we investigate the relationships among rough sets, soft sets and hemirings. The concept of soft rough hemirings is introduced, which is an extended notion of a rough hemiring. It is pointed out that in this paper, we first apply soft rough sets to algebraic structure-hemirings. Further, we first put forward the concepts of C-soft sets and CC-soft sets, which provide a new research idea for soft rough algebraic research. Moreover, we study roughness in hemirings with respect to MSR-approximation spaces. Some new soft rough operations over hemirings are explored. In particular, lower and upper MSR-hemirings (k-ideal and h-ideal) are investigated. Finally, we put forth an approach for multicriteria group decision making problem based on modified soft rough sets and offer an actual example.     

EPCCL理论的求交知识编译算法

超扩展规则是对扩展规则的扩充,基于超扩展规则能够求得任意两个非互补且不相互蕴含的子句所能扩展出极大项集的交集、差集和并集,并将所得结果以EPCCL（each pair of clauses contains complementary literals）理论的形式保存.基于超扩展规则的性质,本文提出了一种新的EPCCL理论编译算法：求交知识编译算法IKCHER（intersection approach to knowledge compilation based on hyper extension rule）,该算法适合难解类SAT问题的知识编译,同时是一种可并行的知识编译算法.本文还研究了如何实现多个EPCCL理论的求交操作,证明了EPCCL理论的求交过程是可并行执行的,并设计了相应并行求交算法PIAE（parellel intersection of any number of EPCCL）.通过对输入EPCCL理论对应普通子句集的利用,设计了一种高效的并行求交算法imp-PIAE（improvement of PIAE）.基于上述算法本文还设计了两个并行知识编译算法P-IKCHER（IKCHER with PIAE）和impP-IKCHER（IKCHER withimp-PIAE）,分别采用PIAE并行合并算法和imp-PUAE并行合并算法.最后,通过实验验证了大部分情况下IKCHER算法的编译质量是目前为止所有EPCCL理论编译器中最优的,P-IKCHER算法所使用的合并策略并没有起到加速的效果,反而使得编译效率和编译质量有所下降,而impP-IKCHER算法提高了IKCHER算法的编译效率,四核并行下最高可提高两倍.