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 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.     

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.     

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 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.     

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 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 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.     

A characterization of fuzzy subgroups of some Abelian groups

We introduce the notion of fuzzy p^*-subgroups and characterize fuzzy subgroups of torsion groups and cyclic groups by their minimal fuzzy p-subgroups and minimal fuzzy p^*-subgroups.     

Fuzzy logic and the calculi of fuzzy rules, fuzzy graphs, and fuzzy probabilities

The past few years have witnessed a rapid growth in the number and variety of applications of fuzzy logic, ranging from consumer products and industrial process control to medical instrumentation, information systems, and decision analysis. The foundations of fuzzy logic have become firmer and its impact within the basic sciences—and especially in mathematical and physical sciences—has become more visible and more substantive. And yet, there are still many misconceptions about the aims of fuzzy logic and misjudgments of its strengths and limitations.One of the common misconceptions is rooted in semantics: as a label, fuzzy logic, FL, has two different meanings. More specifically, in a narrow sense, fuzzy logic, FLn, is a logical system which aims at a formalization of approximate reasoning. In this sense, fuzzy logic is an extension of multivalued logic but its agenda is quite different from that of conventional multivalued systems.In a wide sense, fuzzy logic, FLw, is coextensive with fuzzy set theory, FST. Flw is far broader than FLn and contains Fln as one of its branches. Today, the term fuzzy logic is used predominantly in its wide sense. Thus, effectively, FL = FLw = FST.Another important point is that any field X can be fuzzified by replacing crisp sets in X by fuzzy sets. For example, through fuzzification, arithmetic can be generalized to fuzzy arithmetic, topology to fuzzy topology, control theory to fuzzy control theory, etc.In this perspective, the calculi of fuzzy rules (CFR), fuzzy graphs (CFG), and fuzzy probabilities (CFP) may be viewed as generalizations of the calculi of rules, graphs, and probabilities. The importance of CFR, CFG, and CFP derives from the fact that they play a central role in most of the applications of FL. In particular, the calculus of fuzzy graphs, which is a subset of the calculus of fuzzy rules, accounts for most of the applications of fuzzy logic in control, systems analysis, and related fields.Central to the calculus of fuzzy rules is a language referred to as the Fuzzy Dependency and Command Language (FDCL). The syntax of FDCL is concerned with the form of rules, while the semantics of FDCL is concerned with their meaning. An important issue in CFR is that of the induction of rules from observations.In CFG, a fuzzy graph is defined as the disjunction of Cartesian products of fuzzy sets. In effect, a fuzzy graph may be viewed as a compressed representation of a functional or relational dependence. Operations on fuzzy graphs play an important role in CFG.In the calculus of fuzzy probabilities, probabilities are assumed to be represented as fuzzy rather than crisp numbers. In a related way, probability distributions are represented as fuzzy graphs. A major aim of CFP is to provide a framework for linguistic decision analysis—a type of qualitative analysis in which fuzzy numbers and fuzzy graphs are employed to represent both probabilities and utilities.In an essential way, the methodologies of fuzzy rules, fuzzy graphs, and fuzzy probabilities reflect the fact that

1. (a) imprecision and uncertainty are pervasive; and

2. (b) precision and certainty carry a cost.

In the final analysis, the principal aim of these methodologies is to exploit the tolerance for imprecision and uncertainty to achieve tractability, robustness, and low solution cost.     

Fuzzy automata with fuzzy relief

This paper shows a definition of a fuzzy automaton, which has the state, input, and output sets as fuzzy sets. The state transition function is defined as moving on a fuzzy relief with fuzzy peak-states and boundaries between different membership functions. After the definition of fuzzy automaton with fuzzy relief, the paper deals with a generalization, simulation and realization of such a fuzzy automaton. The paper links the defined fuzzy automaton to an existing fuzzy JK memory cell and to well-known fuzzy automata defined on the basis of crisp sets     

Fuzzy control with fuzzy inputs

This paper is concerned with the use of fuzzy inputs in fuzzy logic controllers. A precise representation of fuzzy logic controllers by means of mappings is used to introduce different ways for dealing with fuzzy inputs. Two types of fuzzy inputs are presented and their potential use in fuzzy control is discussed. The proposed concepts are applied to control a first order process with a PI controller. This simple process is chosen to clearly illustrate the behavior of the closed-loop system using fuzzy inputs for fuzzy reference and fuzzy measurement. Finally, a nonlinear process is used to illustrate the effects of fuzzy inputs on a more complex system. Although it is sometimes speculated that fuzzy inputs may improve the behavior of fuzzy controllers, experiments developed in this paper show this point is not straightforward and that the relevance of fuzzy inputs should be questioned in closed-loop fuzzy control.     

Fuzzy semi-parametric partially linear model with fuzzy inputs and fuzzy outputs

A large number of accounting studies have focused on parametric or non-parametric forms of fuzzy regression relationships between dependent and independent variables. Notably, semi-parametric partially linear model as a powerful tool to incorporate statistical parametric and non-parametric regression analyses has gained attentions in many real-life applications recently. However, fuzzy data find application in many real studies. This study is an investigation of semi-parametric partially linear model for such cases to improve the conventional fuzzy linear regression models with fuzzy inputs, fuzzy outputs, fuzzy smooth function and non-fuzzy coefficients. For this purpose, a hybrid procedure is suggested based on curve fitting methods and least absolutes deviations to estimate the fuzzy smooth function and fuzzy coefficients. The proposed method is also examined to be compared with a common fuzzy linear regression model via a simulation data set and some real fuzzy data sets. It is shown that the proposed fuzzy regression model performs more convenient and efficient results in regard to six goodness-of-fit criteria which concludes that the proposed model could be a rational substituted model of some common fuzzy regression models in many practical studies of fuzzy regression model in expert and intelligent systems.