THE THEORY OF GLOBAL FUZZY NEIGHBORHOOD STRUCTURES - PART I - THE GENERAL-CASE

This paper is devoted to global fuzzy neighborhood structures. We intr oduce three types of these structures, defined by means of fuzzy filte rs. 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, hom ogeneous fuzzy filters are used which are representable by a prefilter , or equivalently, prefilters are used which represent special homogen eous fuzzy filters. All fuzzy topologies and stratified fuzzy topologi es are global fuzzy neighborhood structures of the first and second ty pe, respectively. They appear in a canonical way as interior operators . Fuzzy neighborhood structures introduced by Lowen [Fuzzy Sets and Sy stems 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 ca n be identified with fuzzy topologies, However, the related fuzzy topo logical approach differs. Fuzzy neighborhood structures in sense of Lo wen are characterized canonically as fuzzy closure operators. In this paper the relations between the three types of global fuzzy neighborho od structures and their associated fuzzy topologies and also some rela tions to the fuzzy neighborhood structures in the sense of Lowen are i nvestigated. Moreover, this paper deals with initial and final structu res of global fuzzy neighborhood structures. In two subsequent papers (Part II and Part III), fuzzy topogenous orders and fuzzy uniform stru ctures will be investigated, respectively. All regular fuzzy topogenou s orders, that is, all fuzzy topogenous structures, and in particular, all fuzzy proximities, are global fuzzy neighborhood structures. As i s shown by examples there exist global fuzzy neighborhood structures w hich are not fuzzy topogenous structures. Hence, the notion of global fuzzy neighborhood structure is more rich. Fuzzy uniform structures, d efined analogously to A. Weil's definition of a uniform structure as f uzzy filters, generate in a canonical way global fuzzy neighborhood st ructures. Some results on global fuzzy neighborhood structures will al so be published in Gahler et al. (submitted), in particular those rela ted to stratifications. (C) 1998 Elsevier Science B.V. All rights rese rved.