双极值模糊软子群和双极值模糊正规软子群

研究了双极值模糊软子群的等价刻画。在双极值模糊软子群的基础上定义了双极值模糊正规软子群,得到了它的一些性质及等价刻画,进一步还研究了在双极值模糊软同态下,双极值模糊正规软子群的像与原像一些性质。     

Preferential normal fuzzy subgroups

In this paper we consider the notion of normality on preferential fuzzy subgroups of a finite group G and define the associated concept of a normal pinned-flag. We discuss the preferential equality of quotients and products of normal preferential fuzzy subgroups. Further normalizers of fuzzy subgroups under preferential equality are briefly dealt with. Examples are given to illustrate the structure of preferential normal fuzzy subgroups and normal pinned-flags.     

Fuzzy orders relative to fuzzy subgroups

In this paper, we introduce the notion of fuzzy orders of the elements of a group and show that most of the basic properties of orders of elements in the group theory are valid in the theory of fuzzy groups where ordinary orders of elements are replaced by fuzzy orders of elements. Reports of further development of the subject will follow in subsequent essays.     

Fuzzy moves using normal fuzzy reasoning with fuzzy knowledge discovery

By merging the theory of moves (TOM) and the fuzzy sets theory, we have developed the theory of fuzzy moves (TFM) to make better fuzzy moves for fuzzy games. Since the data granularity of conventionally used fuzzy sets is too low to contain more heuristic information and mined knowledge, we take primary fuzzy sets with higher data granularity as fundamental elements for fuzzy reasoning so as to make more reasonable moves. The simulation results indicate that (1) TFM with normal fuzzy reasoning can make better and more reasonable moves than TOM with precise reasoning since different global strategies are taken into account by TFM and (2) the novel fuzzy reasoning methodology is more reasonable and more useful to make fuzzy moves than the conventional one. ©1999 John Wiley & Sons, Inc.     

群上的模糊软同余

为了进一步研究模糊软集理论,通过将软集与模糊同余结合,给出了模糊软同余的定义,并在一定程度上推广了模糊软集。研究了群上的模糊软同余的相关性质。建立了群上的模糊软同余与正规模糊软群之间的联系,给出了软模糊同态定理。     

Fuzzy subgroups and similarities

 Given a set S, we show that there is a strict relation between the notion of similarity on S and the one of fuzzy subgroup of transformations in S . Such a relation enables us to extablish a connection between fuzzy subgroups and distances.     

Some constructions of the join of fuzzy subgroups and certain lattices of fuzzy subgroups with sup property

In this paper, some new lattices of fuzzy substructures are constructed. For a given fuzzy set μ in a group G, a fuzzy subgroup S(μ) generated by μ is defined which helps to establish that the set L_s of all fuzzy subgroups with sup property constitutes a lattice. Consequently, many other sublattices of the lattice L of all fuzzy subgroups of G like , etc. are also obtained. The notion of infimum is used to construct a fuzzy subgroup i(μ) generated by a given fuzzy set μ, in contrast to the usual practice of using supremum. In the process a new fuzzy subgroup i(μ)^∗ is defined which we shall call a shadow fuzzy subgroup of μ. It is established that if μ has inf property, then i(μ)^∗ also has this property.     

The schnittaxiom and unions of fuzzy subgroups

A necessary and sufficient condition for the union of an arbitrary family of fuzzy subgroups of a group to be a fuzzy group has been proposed. The criterion which gives a cut of the unit interval is called the Schnittaxiom. It has been applied successfully to investigate fuzzy subgroupness of arbitrary unions of homomorphic images and preimages.     

Distributivity in lattices of fuzzy subgroups

The main goal of this paper is to study the finite groups whose lattices of fuzzy subgroups are distributive. We obtain a characterization of these groups which is similar to a well-known result of group theory.     

Fuzzy subgroups having the property

We introduce the notion of the property ( * ) for a fuzzy group and characterize all finite cyclic groups in terms of this notion. This property concerns the transition of the order of an element of a group from the classical to the fuzzy setting.     

双极值模糊（反）软子群

在双极值模糊软集理论的基础上,给出了双极值模糊（反）软子群的概念,讨论了它们的一些相关性质及等价刻画。提出了双极值模糊软映射下双极值模糊软集的像与原像的概念,并研究了双极值模糊软同态下双极值模糊（反）软子群的同态像与原像的初等性质。     

Fuzzy relations and fuzzy groups

Given a group S, we consider fuzzy relations on S, that is, maps from S × S into [0,1]. Of particular interest is to investigate conditions under which the fuzzy relation becomes a fuzzy subgroup on S × S. We prove that if σ is a fuzzy subset of S and μ_σ is the strongest fuzzy relation on S that is a fuzzy relation on σ, then μ_σ is a fuzzy subgroup if and only if σ is a fuzzy subgroup. A number of other results are obtained about the interrelationships between fuzzy relations on S (including the weakest fuzzy relation) and fuzzy subgroups on S × S.     

Fuzzy C-means and fuzzy swarm for fuzzy clustering problem

Fuzzy clustering is an important problem which is the subject of active research in several real-world applications. Fuzzy c-means (FCM) algorithm is one of the most popular fuzzy clustering techniques because it is efficient, straightforward, and easy to implement. However, FCM is sensitive to initialization and is easily trapped in local optima. Particle swarm optimization (PSO) is a stochastic global optimization tool which is used in many optimization problems. In this paper, a hybrid fuzzy clustering method based on FCM and fuzzy PSO (FPSO) is proposed which make use of the merits of both algorithms. Experimental results show that our proposed method is efficient and can reveal encouraging results.     

Fuzzy subgroups: Some characterizations. II

We consider an analysis of fuzzy subgroups in terms of the corresponding family of level subgroups. Given a finite chain of subgroups of a group G, we prove that there exists a fuzzy subgroup of G whose level subgroups are exactly the subgroups of this chain. As a corollary we obtain an interpretation of the number of chains of subgroups of a group G in which a subgroup H is a member. When the group G is a supersolvable group, some further interpretation of this number is obtained.     

On the definition of the intuitionistic fuzzy subgroups

In this paper, a new kind of intuitionistic fuzzy subgroup theory, which is different from that of Ma, Zhan and Davvaz (2008) [22], [23], is presented. First, based on the concept of cut sets on intuitionistic fuzzy sets, we establish the neighborhood relations between a fuzzy point $x_a$ and an intuitionistic fuzzy set $A$. Then we give the definitions of the grades of $x_a$ belonging to $A$, $x_a$ quasi-coincident with $A$, $x_a$ belonging to and quasi-coincident with $A$ and $x_a$ belonging to or quasi-coincident with $A$, respectively. Second, by applying the 3-valued Lukasiewicz implication, we give the definition of $(\alpha ,\beta )$-intuitionistic fuzzy subgroups of a group $G$ for $\alpha ,\beta \in \{\in ,q,\in \wedge q,\in \vee q\}$, and we show that, in 16 kinds of $(\alpha ,\beta )$-intuitionistic fuzzy subgroups, the significant ones are the $(\in ,\in )$-intuitionistic fuzzy subgroup, the $(\in ,\in \vee q)$-intuitionistic fuzzy subgroup and the $(\in \wedge q,\in )$-intuitionistic fuzzy subgroup. We also show that $A$ is a $(\in ,\in )$-intuitionistic fuzzy subgroup of $G$ if and only if, for any $a\in (0,1]$, the cut set $A_a$ of $A$ is a 3-valued fuzzy subgroup of $G$, and $A$ is a $(\in ,\in \vee q)$-intuitionistic fuzzy subgroup (or $(\in ,\in \vee q)$-intuitionistic fuzzy subgroup) of $G$ if and only if, for any $a\in (0,0.5]$(or for any $a\in (0.5,1]$), the cut set $A_a$ of $A$ is a 3-valued fuzzy subgroup of $G$. At last, we generalize the $(\in ,\in )$-intuitionistic fuzzy subgroup, $(\in ,\in \vee q)$-intuitionistic fuzzy subgroup and $(\in \wedge q,\in )$-intuitionistic fuzzy subgroup to intuitionistic fuzzy subgroups with thresholds, i.e., $(s,t]$-intuitionistic fuzzy subgroups. We show that $A$ is a $(s,t]$-intuitionistic fuzzy subgroup of $G$ if and only if, for any $a\in (s,t]$, the cut set $A_a$ of $A$ is a 3-valued fuzzy subgroup of $G$. We also characterize the $(s,t]$-intuitionistic fuzzy subgroup by the neighborhood relations between a fuzzy point $x_a$ and an intuitionistic fuzzy set $A$.     

Fuzzy inference and fuzzy inference processor

Fuzzy inference, a data processing method based on the fuzzy theory that has found wide use in the control field, is reviewed. Consumer electronics, which accounts for most current applications of this concept, does not require very high speeds. Although software running on a conventional microprocessor can perform these inferences, high-speed control applications require much greater speeds. A fuzzy inference date processor that operates at 200000 fuzzy logic inferences per second and features 12-b input and 16-b output resolution is described     

Fuzzy relations and fuzzy relational databases

This paper deals with the connections existing between fuzzy set theory and fuzzy relational databases. Our new result dealing with fuzzy relations is how to calculate the greatest lower bound (glb) of two similarity relations. Our main contributions in fuzzy relational databases are establishing from fuzzy set theory what a fuzzy relational database should be (the result is both surprising and elegant), and making fuzzy relational databases even more robust.Our work in fuzzy relations and in fuzzy databases had led us into other interesting problems—two of which we mention in this paper. The first is primarily mathematical, and the second provides yet another connection between fuzzy set theory and artificial intelligence. In understanding similarity relations in terms of other fuzzy relations and in making fuzzy databases more robust, we work with closure and interior operators; we present some important properties of these operators. In establishing the connection between fuzzy set theory and artificial intelligence, we show that an abstraction on a set is in fact a partition on the set; that is, an abstraction defines an equivalence relation on the underlying set.     

Fuzzy operator logic and fuzzy resolution

There have been only few attempts to extend fuzzy logic to automated theorem proving. In particular, the applicability of the resolution principle to fuzzy logic has been little examined. The approaches that have been suggested in the literature, however, have made some semantic assumptions which resulted in limitations and inflexibilities of the inference mechanism. In this paper we present a new approach to fuzzy logic and reasoning under uncertainty using the resolution principle based on a new operator, the fuzzy operator. We present the fuzzy resolution principle for this logic and show its completeness as an inference rule.     

Fuzzy fractals and fuzzy turbulence.

In this paper, we have defined and discussed fuzzy fractals from image generation point of view. We have also proposed a fuzzy system modeling of a two dimensional turbulence just as a chaotic occurrence of fuzzy vortices in a two dimensional dynamic fluid.