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

Abstract

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.