Decomposition of fuzzy continuity and fuzzy ideal continuity via fuzzy idealization

Recently, El-Naschie has shown that the notion of fuzzy topology may be relevant to quantum paretical physics in connection with string theory and E-infinity space time theory. In this paper, we study the concepts of r-fuzzy semi-I-open, r-fuzzy pre-I-open, r-fuzzy α-I-open and r-fuzzy β-I-open sets, which is properly placed between r-fuzzy openness and r-fuzzy α-I-openness (r-fuzzy pre-I-openness) sets regardless the fuzzy ideal topological space in Ŝostak sense. Moreover, we give a decomposition of fuzzy continuity, fuzzy ideal continuity and fuzzy ideal α-continuity, and obtain several characterization and some properties of these functions. Also, we investigate their relationship with other types of function.     

On fuzzy cliques in fuzzy networks

A definition of fuzzy clique in social networks is suggested which overcomes five limitations of current definitions. This definition is based on the networks in which the 0–1 strengths, the weighted strengths, and fuzzy strengths are all allowed. The fuzzy distance in such a network is defined. The node‐clique and clique‐clique coefficients are suggested. The core and the periphery of fuzzy cliques are discussed formally. A “cone like” property of the cores is discovered. The network structures are discussed using the new definition. A “no circle” property of networks is found. Basic fuzzy tools and the related algorithms are also discussed. Some examples are analyzed to demonstrate the theory.     

Eigen fuzzy sets and fuzzy relations

When R is a fuzzy relation between the elements of a finite set X, the fuzzy sets A of X such that R ° A = A (MAX-MIN composition) are called eigen fuzzy sets. The main result of this paper is the determination of the greatest eigen fuzzy set associated with a given fuzzy relation and we give three methods illustrated by an example. We then state that the greatest eigen fuzzy set associated with $\text{R}\text{?}$, the transitive closure of R, is exactly the one associated with R. Finally we describe how to obtain all fuzzy relations keeping invariant a given fuzzy set.     

Transitive fuzzy orderings of fuzzy numbers

In this paper we introduce a generalization of the Baas-Kwakernaak index by replacing the min operation in their definition by a t-norm. Some properties of the thus defined induced fuzzy ordering are established. In particular, it is shown that restrictions of the induced fuzzy ordering on some special classes of fuzzy numbers are reflexive fuzzy orders.     

Complete fuzzy topological hyperfields and fuzzy multiplication in the fuzzy real lines

Recently E. Sanchez has given a necessary and sufficient condition for the existence of a solution of a fuzzy $\text{A 1 X = C}$, with 1 the extension of a given binary operation in the universe of discourse.Using many valued logic we generalize this result and get a characterization also of the cases if $\text{A 1 X = C}$ has only nearly a solution.Additionally, we consider fuzzy relation equations and get similar results.     

Folding of fuzzy torus and fuzzy graphs

This paper is a continuation of El-Ghoul (1993). It introduces a new definition for the folding of fuzzy torus. It gives the concept of the connection between the folding of fuzzy graphs and the folding of fuzzy torus. Theorems governing this connection are established here.     

Good fuzzy preorders on fuzzy power structures

This paper deals with good fuzzy preorders on fuzzy power structures. It is shown that a fuzzy preorder R on an algebra (X,\mathbbF){(X,\mathbb{F})} is compatible if and only if it is Hoare good, if and only if it is Smyth good.     

Fuzzy topology on fuzzy sets: Product fuzzy topology and fuzzy topological groups

The notion of product fuzzy topology in the case of fuzzy topology on fuzzy sets is introduced and the product invariance of fuzzy Hausdorffness, compactness, connectedness are examined. The product fuzzy topology is used to define fuzzy group topology on a fuzzy subgroup of a group G and some properties of fuzzy topological groups are obtained.     

Dynamic programming for fuzzy systems with fuzzy environment

A dynamic programming functional equation for a multi-stage decision problem with fuzzy dynamics and environment is formulated and solved by a process of fuzzy interpolation. This is an extension of the Bellman-Zadeh model.     

A new fuzzy compactness defined by fuzzy nets

Let (Ω, $\text{A}$, μ) be a probability space and let $\text{B}$ be a subsigma algebra of $\text{A}$. Let A= L_∞Ω, $\text{A}$, μ , let A= L_∞Ω, $\text{B}$, μ, and let f?A. It is shown that best L_∞-approximations of f by elements of B comprise an interval in B; that is, there exists $\text{f}\text{,}\text{f}\text{?B}$ such that a function g?B is a best L_∞-approximation to f if and only if $\text{f}\text{? g ?}\text{f}$ a.e. on Ω. The difference, $\text{f}\text{? f}$, of $\text{f}$ and f is completely characterized in terms of special sets that have been developed in [2]. Then it is established that the best best L_∞-approximation, f_{$\text{B}$,∞}, to f by elements of B is the average of $\text{f}$ and $\text{f}$, where the function f_{$\text{B}$,∞} is defined by f_{$\text{B}$,∞}(ω) lim_{p → ∞}f_$\text{B}$,P(ξ) and f_$\text{B}$,P denotes the best L_p-approximation to f elements of L_p(Ω, $\text{B}$, μ).     

On fuzzy ideals and fuzzy bi-ideals in semigroups

In this paper we give some properties of fuzzy ideals and fuzzy bi-ideals of semigroups, and characterize semigroups that are (left) duo, (left) simple and semilattices of subsemigroups in terms of fuzzy ideals and fuzzy bi-ideals.     

The fuzzy core in games with fuzzy coalitions

In this paper, the fuzzy core of games with fuzzy coalition is proposed, which can be regarded as the generalization of crisp core. The fuzzy core is based on the assumption that the total worth of a fuzzy coalition will be allocated to the players whose participation rate is larger than zero. The nonempty condition of the fuzzy core is given based on the fuzzy convexity. Three kinds of special fuzzy cores in games with fuzzy coalition are studied, and the explicit fuzzy core represented by the crisp core is also given. Because the fuzzy Shapley value had been proposed as a kind of solution for the fuzzy games, the relationship between fuzzy core and the fuzzy Shapley function is also shown. Surprisingly, the relationship between fuzzy core and the fuzzy Shapley value does coincide, as in the classical case.     

A study of normal fuzzy subgroups and characteristic fuzzy subgroups of a fuzzy group

In this paper, we investigate the properties of normal fuzzy subgroups of a fuzzy group. We also introduce the notion of a characteristic fuzzy subgroup of a fuzzy group, of which we provide level subset and strong level subset characterizations. Then we prove that a characteristic fuzzy subgroup of a fuzzy group is a normal fuzzy subgroup. Besides we prove that the commutator subgroup generated by the commutator of two normal fuzzy subgroups of a fuzzy group is contained in their intersection. Finally, we construct many new lattices and sublattices of fuzzy subgroups and normal fuzzy subgroups of a given fuzzy group. We also construct lattices of characteristic fuzzy subgroups possessing sup-property.