Fuzzy Uniform Structures Research Articles

Completeness of Hutton $[0,1]$-quasi-uniformities induced by a crisp uniformity

In cite{Kats84}, Katsaras introduced a method for constructing a Hu-tton $[0,1]$-quasi-uniformity from a crisp uniformity. In this paper we present other different methods for making this based mainly in the concept of a fuzzy uniform structure. Furthermore, we prove that some of these methods preserve the completeness property of the quasi-uniformity. Moreover, we also show that Katsaras' construction allows to develop a theory of completion of a Hu-tton [0,1]-quasi-uniform space obtained from a uniform space.

English

English

Fuzzy uniform structures

The concept of fuzzy uniform structure was introduced in [7] as a fuzzy counterpart of the concept of gauge associated with a uniformity. In fact, the category of fuzzy uniform structures is isomorphic to that of uniform spaces. Here, we introduce two other concepts of fuzzy uniform structures which allow to establish two categories isomorphic to the categories of probabilistic uniform spaces and Lowen uniform spaces, respectively. This sheds light on the relationship between these fuzzy uniformities and classical uniformities. Furthermore, we obtain a factorization of Lowen?s adjoint functors ?* and ? which establish a relationship between the categories of uniform spaces and Lowen uniform spaces.

Standard fuzzy uniform structures based on continuous t-norms

This paper deals with fuzzy uniform structures previously introduced by the authors [Fuzzy uniform structures and continuous t-norms, Fuzzy Sets Syst. 161 (2009) 1011–1021]. Our approach involves a covariant functor Ψ from the category of fuzzy uniform spaces and fuzzy uniformly continuous mappings (in our sense) to the category of uniform spaces and uniformly continuous mappings. We show that Ψ is well-behaved with respect to some significant fuzzy uniform concepts, and its behavior provides a method to introduce notions of fine fuzzy uniform structure and Stone–Čech fuzzy compactification in this context. Our method also applies to obtain fuzzy versions of some classical results on topological algebra and hyperspaces. The case of quasi-uniform structures is also analyzed.

Fuzzy uniform structures and continuous t-norms

We introduce and discuss a notion of fuzzy uniform structure that provides a direct link with the classical theory of uniform spaces. More exactly, for each continuous t-norm we prove that the category of all fuzzy uniform spaces in our sense (and fuzzy uniformly continuous mappings) is isomorphic to the category of uniform spaces (and uniformly continuous mappings) by means of a covariant functor. Moreover, we also describe the inverse functor and, then, we discuss completeness and completion of these fuzzy uniform structures. It follows from our results that each fuzzy uniform structure in our sense induces a Hutton [0, 1]-uniformity and, conversely, each Hutton [0, 1]-uniformity induces a fuzzy uniform structure in our sense.

On initial and final fuzzy uniform structures, Part II

This paper is the second part and continuation of a paper for the author published in 2003 and investigating initial fuzzy uniform structures. The final fuzzy uniform structures and the final global fuzzy neighborhood structures, for the notions of fuzzy uniform structure and of global fuzzy neighborhood structure introduced by the author and others in 1998 in two separate papers, are characterized. This paper also shows that the expected relations between the final fuzzy uniform structures and the final fuzzy topologies and the final global fuzzy neighborhood structures are indeed true.

The relation between GTi-spaces and fuzzy proximity spaces, G-compact spaces, fuzzy uniform spaces

In this paper, we study the relation between the fuzzy separation axioms, which had been introduced by the authors in 2001, and the fuzzy proximity defined by Katsaras in 1980. We study also the relation between our fuzzy separation axioms and the G-compactness defined by Gähler in 1995. Moreover, we show here the relation between these fuzzy separation axioms and the fuzzy uniform structures introduced and studied by Gähler and the first author in 1998.

On initial and final fuzzy uniform structures

This paper is devoted to study the initial and final fuzzy uniform structures of the fuzzy uniform structure defined by the author and others in 1998. In this paper, we show that all initial and final lifts in the category FUN of these fuzzy uniform spaces and hence the initial and final fuzzy uniform structures exist. We introduce a characterization for the initial fuzzy uniform structures and as a special initial fuzzy uniform spaces the subspaces and product spaces of these fuzzy uniform spaces can be also characterized. We also show that the fuzzy topology (global homogeneous fuzzy neighborhood structure) associated to the initial fuzzy uniform structure of a family of fuzzy uniform structures coincides with the initial fuzzy topology (initial global homogeneous fuzzy neighborhood structure) of the family of fuzzy topologies (global homogeneous fuzzy neighborhood structures) associated to these fuzzy uniform structures.

The α-levels of a fuzzy uniform structure and of a fuzzy proximity

There are different notions of fuzzy uniform structures and of fuzzy proximities that have been introduced in the literature. In this paper we are interested in the fuzzy uniform structure U in the sense of Gähler et al. (1998) which is defined as some fuzzy filter and we are also interested in the fuzzy proximity N in the sense of Gähler et al. (1998), called the fuzzy proximity of the internal type that is defined by means of another notion of symmetry not depending on an order-reversing involution. Here, we introduce the α-level uniform structure U α and the α-level proximity N α U and N , respectively. Weshow that there is one-to-one correspondence between a fuzzyuniform structure U and the family ( U α ) α∈ L 0 of uniform structuresthat fulfills certain conditions, is given by: U α= U α and U(U)=⋁ A∈U α,χA⩽u α . We also show that the topologies T u α and T N α associated with U α and N α coincides with the α-level topologies of the fuzzy topologies τ U and τ N associated to U and N ,respectively, that is, T U α =(τ U ) α and T N α =(τ N ) α . Moreover, we assign for each fuzzyuniform structure U an associated fuzzy proximity of theinternal type N U and hence we get the relationbetween the α-levels of U and of N U which is given by: N U α =( N U ) α .

The theory of global fuzzy neighborhood structures (III). Fuzzy uniform structures

In this paper a new notion of fuzzy uniform structure is presented and investigated. Fuzzy uniform structures are given as special fuzzy filters. Of special interest are those fuzzy uniform structures which are sup-fuzzy filters. They can be characterized by a related notion of fuzzy syntopogenous structure. As is shown, fuzzy metrics (Gähler and Gähler, 1994) generate fuzzy uniform structures in our sense. Each fuzzy uniform structure has an associated global homogeneous fuzzy neighborhood structure and hence also an associated fuzzy topology. In case of a 0, 1-fuzzy uniform structure and even in a more general case, a fuzzy uniform structure of the first type can be associated. There are given relation to Huttons's and Katsaras' notions of fuzzy uniform structure (Hutton, 1977; Katsaras, 1979) and to the notion of fuzzy syntopogenous structure in sense of Katsaras and Petalas (1984).

The theory of global fuzzy neighborhood structures Part I — The general case

This paper is devoted to global fuzzy neighborhood structures. We introduce three types of these structures, defined by means of fuzzy filters. In some sense, the first type is more general than the second one, and the second type is more general than the third one. In the second case, only homogeneous fuzzy filters are used. In the third case, homogeneous fuzzy filters are used which are representable by a prefilter, or equivalently, prefilters are used which represent special homogeneous fuzzy filters. All fuzzy topologies and stratified fuzzy topologies are global fuzzy neighborhood structures of the first and second type, respectively. They appear in a canonical way as interior operators. Fuzzy neighborhood structures introduced by Lowen [Fuzzy Sets and Systems 7 (1982) 165] are defined by means of prefilters. The definition of these structures is in some sense similar to a characterization of those global fuzzy neighborhood structures of the third type which can be identified with fuzzy topologies. However, the related fuzzy topological approach differs. Fuzzy neighborhood structures in sense of Lowen are characterized canonically as fuzzy closure operators. In this paper the relations between the three types of global fuzzy neighborhood structures and their associated fuzzy topologies and also some relations to the fuzzy neighborhood structures in the sense of Lowen are investigated. Moreover, this paper deals with initial and final structures of global fuzzy neighborhood structures. In two subsequent papers (Part II and Part III), fuzzy topogenous orders and fuzzy uniform structures will be investigated, respectively. All regular fuzzy topogenous orders, that is, all fuzzy topogenous structures, and in particular, all fuzzy proximities, are global fuzzy neighborhood structures. As is shown by examples there exist global fuzzy neighborhood structures which are not fuzzy topogenous structures. Hence, the notion of global fuzzy neighborhood structure is more rich. Fuzzy uniform structures, defined analogously to A. Weil's definition of a uniform structure as fuzzy filters, generate in a canonical way global fuzzy neighborhood structures. Some results on global fuzzy neighborhood structures will also be published in Gähler et al. (submitted), in particular those related to stratifications.

Hyperspace fuzzy binary relations

Abstract We associate with each implication operator in [0, 1]-valued logic, under certain conditions, an algorithm for extending a fuzzy (or ordinary) binary relation φ from X to Y , to a fuzzy binary relation from I X to I Y , said to be a fuzzy hyperspace extension of φ. We show that our extension algorithms preserve a number of properties of binary relations. As application to fuzzy topology, we show that each algorithm produces its notion of Hausdorff fuzzy T -uniformities, and related notions. These subsume Lowen's definition of horizontal hyperspace fuzzy uniform structures, as a special case corresponding to the reciprocal of the Godel implication operator.

Fuzzy nearness structure

Abstract In this paper the notion of fuzzy nearness structure is introduced and some investigations are made towards the unification of several structures: namely, fuzzy topological structure, fuzzy proximity structure, fuzzy contiguity structure and fuzzy uniform structure.