Fuzzy Real Line Research Articles

Fuzzifying pseudo-quasi-metric topologies on the fuzzy real line

In this paper, three natural fuzzifying topologies are presented on the fuzzy real line. Then the notion of fuzzifying pseudo-quasi-metrics is introduced. It is proved that the three fuzzifying topologies can be induced respectively by three fuzzifying pseudo-quasi-metrics. Our definition of fuzzifying pseudo-metric is slightly different from that of KM-fuzzy metric. A fuzzifying pseudo-metrics can be regarded as a weak form of a KM fuzzy metric.

The Fuzzification of Classical Structures: A General View

The aim of this survey article, dedicated to the 50th anniversary of Zadeh’s pioneering paper "Fuzzy Sets" (1965), is to offer a unitary view to some important spaces in fuzzy mathematics: fuzzy real line, fuzzy topological spaces, fuzzy metric spaces, fuzzy topological vector spaces, fuzzy normed linear spaces. We believe that this paper will be a support for future research in this field.

Overview and comparison of localic and fixed-basis topological products

This paper studies localic products, traditional topological products, and L-topological products, and gives a complete outline of the localic product. Comparisons of localic and L-topological products are generally absent in the literature, and this paper answers longstanding open questions in that area as well as provides a complete proof of the classical comparison theorem for localic and traditional topological products. This paper contributes several L-valued comparison theorems, one of which states: the localic and L-topological products of L-topologies are order isomorphic if and only if the localic product is L-spatial, providing L is itself spatial and the family of L-topological spaces is “prime separated”. These last two conditions always hold in the traditional setting, capturing the traditional comparison theorem as a special case, and the prime separation condition is satisfied by important lattice-valued examples that include the fuzzy real line and the fuzzy unit interval for L any complete Boolean algebra and the alternative fuzzy real line and fuzzy unit interval for L any (semi)frame. Separation conditions help control the “sloppy” behavior of the L-topological product when | L | > 2 , and several separation conditions are studied in this context; and it should be noted that localic products have a point-free version of the “product” separation condition considered in this paper. The traditional comparison theorem is carefully proved both to fill gaps in the extant literature and to motivate the L -valued comparison theorem quoted above and reveal the special role played by cross sums of prime ( L-)open subsets. En route, characterizations are given of prime L-open subsets of certain L-products, which in turn yield characterizations of prime open and irreducible closed subsets of traditional product spaces.

Convergence in the [formula omitted]-fuzzy real line

Infinite sums and sequences in the fuzzy real line R ( I ) have been investigated by Wang. In this paper, a necessary and sufficient condition for a sequence in R ( I ) to be convergent is given in terms of α -cut set. It is proved that fuzzy convergence of sequences in R ( I ) is equivalent to weak convergence. As an application, convergence of sequences of fuzzy numbers in the sense of Chang and Zadeh is discussed.

Fixed-point theorems in fuzzy real line

Some sufficient conditions for the existence of fixed points of increasing operators in fuzzy real line R L are given. We also establish some coupled quasi-fixed-point theorems of mixed monotone operators in fuzzy real line R L.

Convergence of sequences on the fuzzy real line

The infinite sum on the fuzzy real line R ( I ) has been investigated by Wang Geping (Fuzzy Sets and Systems 98 (1998) 241). In this paper we introduce the notion of convergence of sequences on R ( I ) , give some conditions for a sequence on R ( I ) to be convergent and discuss some properties of convergence of sequences on R ( I ) .

Separation axioms in [formula omitted]-fuzzy topological spaces (I): [formula omitted] and [formula omitted

Using the notion of remote neighborhood, we define the separation axioms T 0 and T 1 in L-fuzzy topological spaces ( L-fts). The relations between our definitions, Hutton and Reilly's, and Wang's are discussed, and the separations of Hutton's fuzzy unit interval and Gantner's fuzzy real line are examined. Characterizations of these L-fts are proved and a series of properties of them are investigated. Moreover, some results on minimal T 0 L -fts and T 1 L -fts are established, which will be used immediately in our subsequent discussion.

Infinite sum on the fuzzy real line

The fuzzy addition on the fuzzy real line due to Hutton (1975) has been investigated by Lowen (1983), Rodabaugh (1982) and Wang (1988). In this paper we introduce the notion of infinite sums on the fuzzy real line, give a necessary and sufficient condition for an infinite sum to be convergent and discuss some properties of the α-cut of infinite sums.

Some results on [formula omitted] and [formula omitted

This paper gives a partial solution to Problem 1 in Wang and Lu (1992) and a suggestion about how to construct a better fuzzy unit interval. It defines a new kind of fuzzy real line and investigates a series of its properties.

Fuzzy real numbers

The paper gives a contribution to the study of fuzzy real numbers, continuing that in Eklund and Gähler (1988), Gähler and Gähler (1992) and Gähler (1992). We start with investigating algebraic properties using two types of decompositions of fuzzy numbers and the notions of fuzzy norm, of symmetric difference and of supplements of fuzzy numbers. By means of these notions two special fuzzy metrics on the fuzzy real line are given, the fuzzy distance function d and the extended fuzzy distance function d . The main interest concerns fuzzy topological properties of the fuzzy real line. In particular, there are studied the fuzzy topologies T, T d and T d . T is closely related to the fuzzy topology of Hutton's fuzzy unit interval and T d and T d are canonically generated by the fuzzy metrics d and d , respectively.

Fuzzy measurable functions

Abstract A generated fuzzy σ-algebra equipped by Giles fuzzy connectives is considered as a basis of this paper. Starting from a simple fuzzy measurable function, we introduce the notion of a fuzzy measurable function. A one-to-one correspondence between the fuzzy measurable functions and the random variables with values in the fuzzy real line is shown. An inverse of a fuzzy measurable function is proved to be an extended T ∞ -fuzzy observable and vice versa. Linear operations on fuzzy measurable functions are introduced.

Completeness of the standard fuzzy real line for FNS

Abstract We consider a special class of fuzzy metric neighbourhood spaces, where the fuzzy distance is induced by a fuzzy pseudo-norm. This class is contained in Lowen's well known category FNS of Fuzzy Neighbourhood Spaces. We define the fuzzy limit of a sequence, a fuzzy Cauchy sequence and fuzzy convergence in fuzzy metric neighbourhood spaces. We prove that the set of all nonnegative fuzzy real numbers R ∗ (I) under the N-Euclidean metric (the standard fuzzy real line for FNS) is complete, but that the smallest fuzzy neighbourhood real algebra M(I) including R ∗ (I) is not complete. We also characterize the fuzzy Cauchy sequences in M(I) in terms of their corresponding (real) sequences of α-cuts.

Fuzzy metric neighbourhood spaces

Höhle founded the first coherent topological study of Menger's probabilistic metric spaces by endowing them with fuzzy T-uniformities. We continue Höhle's work in the special case when the triangular norm T used is Min, that is when the resulting probabilistic metric uniformities are Lowen fuzzy uniformities. In this case, we call the induced fuzzy topological spaces fuzzy pseudo-metric neighbourhood spaces. We introduce axioms of N-regularity and N-normality, which are satisfied in fuzzy pseudo-metric neighbourhood spaces. We introduced a Lowen fuzzy uniform topology on the fuzzy real line R I, called the N-Euclidean topology, by means of a fuzzy (=probabilistic) metric on R( I), called the axiom of N-complete regularity which characterizes the Lowen fuzzy uniform topologies.

Continuity of fuzzy multiplication in the N-Euclidean space

The N-Euclidean space M(I) is a fuzzy normed vector space, whose underlying set is the smallest real vector space including all nonnegative fuzzy real numbers. The fuzzy multiplication on the fuzzy real line easily extends to a multiplication on M(I) . We show that, under that multiplication, M(I) is a fuzzy normed algebra. In consequence, fuzzy multiplication is shown to be continuous in the N-Euclidean topology on M(I) .

Totally bounded fuzzy syntopogenous structures

Some of the properties of the totally bounded fuzzy syntopogenous structures and the totally bounded fuzzy quasi-uniformities are studied. It is also proved that the restriction to a subset Y, of the fuzzy real line, of the usual fuzzy uniformity is totally bounded iff Y is bounded.

Fuzzy addition in the lowen fuzzy real line

For L a complete chain with order-reversing involution α→ α′, Rodabaugh [7] defined a fuzzy addition ⊛ in the Hutton fuzzy real line R( L) and studied some of its properties. Subsequently, Lowen [3] made a considerably simpler description of ⊛ and gave a direct proof of the continuity of ⊛. In this paper, we introduce the fuzzy addition ⊛ in the Lowen fuzzy real line μ <( R) [6]. We give several equivalent descriptions of ⊛ which may be used for investigating some properties of ⊛. Moreover, we prove the closedness of ⊛ in the four important subspaces A <( R), F <( R), μ < Z ( R), μ < Q ( R) [4] of μ <( R). This implies, in particular, that the sum of two fuzzy rational numbers is fuzzy rational.

Basic notions for fuzzy topology, I

A new notion of fuzzy real line is presented. Applications are obtained when solving the problem of generating fuzzy topologies by fuzzy metrics. The results provide a fundamental example as a basis for a convergence based approach to fuzzy topology.

Dynamic topologies and their applications to crisp topologies, fuzzifications of crisp topologies, and fuzzy topologies on the crisp real line

We introduce and develop topologies whose members are dynamic sets à la Zeleny. This development spawns the notions of phase and co-phase spaces, fuzzy dual, dynamic dual, and co-fuzzy dual with the following results being obtained: Result A. The notion of fuzzy dual provides a unifying framework for the diverse fuzzification schemes of Hutton-Gantner-Steinlage-Warren, Klein, and Lowen. Result B. The dynamic duals of the fuzzy real lines are both homeostatic and coheterostatic. Result C. The fuzzy real lines, re-interpreted as fuzzy duals of R with the usual Euclidean topology T , induce on R , via the notion of co-fuzzy dual, a class of canonical, stratified fuzzy topologies {CO R L }. E.g., if L is a DeMorgan algebra and α ϵ L c -{1}, then the α-compactness of A ⊂ R is equivalent to the ordinary compactness of A w.r.t. T ; and if L is a Hutton algebra, ( R , L, CO R L ) is metrizable in the sense of Hutton and Erceg with Erceg metric d( L) such that d(L) (p r 1, p s 1)=¦r − s¦ (where p r is the crisp point or impulse with support r), i.e., d( L) extends the Euclidean metric on R .

A theory of fuzzy uniformities with applications to the fuzzy real lines

For each completely distributive lattice L with order-reversing involution, the fuzzy real line R (L) is uniformizable by a uniformity which both generates the canonical (fuzzy) topology and induces a pseudometric generating the canonical topology. If L is also a chain, the usual addition and multiplication defined on R  R ({0, 1}) extend jointly (fuzzy) continuously to ⊕ and ⊙ on R (L). Three fundamental questions in fuzzy sets until now are: Question A. If L 1≅ L 2, is R (L 1) uniformly isomorphic to R (L 2) in some sense? Question B. For each chain L, is ⊕ (jointly) uniformly continuous in a sense which guarantees its (joint) continuity on R (L)? Question C. Is R (L) a complete pseudometric space in some sense? We construct categories QU and U using the [quasi-] uniformities of B. Hutton which enable us to answer these questions in the affirmative. These results enhance the canonical standing of the fuzzy real lines and so give additional justification for answering in the affirmative: Question D. Does fuzzy topology have deep, specific, canonical examples?

Connectedness on fuzzy topological spaces

In this paper, properties of connectedness on completely distributive topological lattices (Definition 2.1) and fuzzy topological spaces [1] are studied. Some equivalent propositions on the concepts of functions in completely distributive lattices, and connected elements in completely distributive lattices and fuzzy topological spaces are proved. Specially, connectedness is characterized both by the R-neighborhoods of molecules and C-separations. For any family {( L i , τ i ): i ϵ I} of fuzzy topological spaces, the product [1] ⊗ is connected iff every ( L i , τ i ) is connected. A continuous function maps connected elements onto connected ones. The connectedness on the fuzzy real line R ( L) is discussed in the last section as an important example. The intervals defined here (Definition 4.1) must be connected in the fuzzy real line R ( L), and so the intervals defined by Rodabaugh in [13] are too. Any connected element is connected in the fuzzy real line R ( L), if and only if R ( L) ⋍ R .