Fatma Bayoumi

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 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. Weils 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 Gahler et al. (submitted), in particular those related to stratifications.

The theory of global fuzzy neighborhood structures. II: fuzzy topogenous orders

Abstract This paper deals with fuzzy topogenous orders, in particular, with fuzzy topogenous structures and with the more special fuzzy proximities. These structures have been investigated ay Katsaras and Petalas (1983, 1984). The notion of fuzzy proximity was introduced by Katsaeas (1980). Fuzzy topogenous structures and fuzzy proximities are represented in this paper as global fuzzy neighborhood structures. Fuzzy topogenous orders, in general, are characterized by the notion of global fuzzy neighborhood prestructure, which is a weakening of that one of global fuzzy neighborhood structure. In this paper, moreover, a modification of the notion of fuzzy proximity is considered, called fuzzy proximity of the internal type. Whereas the notion of fuzzy proximity proposed by Katsaras depends on a fixed order-reversing involution of the related lattice L , the notion of fuzzy proximity of the internal type is independent on such an involution. The investigations intthis paper demonstrate that there are important global fuzzy neighborhood structures and prestructures different from fuzzy topologies and fuzzy pretopologies, respectively.

Global L-neighborhood groups

This paper introduces and studies the notion of global L-neighborhood group which is defined as a group equipped with a global L-neighborhood structure in sense of Gahler et al. such that both the binary operation and the unary operation of the inverse are continuous with respect to this global L-neighborhood structure. Some examples of global L-neighborhood groups are given. It is shown that the L-topological groups, given by Ahsanullah in 1984 and later by Bayoumi in 2003, are special global L-neighborhood groups. We also show that all initial and final lifts and hence all initial and final global L-neighborhood groups uniquely exist in the category L-GnghGrp of global L-neighborhood groups. These initial and final global L-neighborhood groups are defined using the initial and final global L-neighborhood structures. Moreover, we show that the L-neighborhood groups, defined by Ahsanullah using the L-neighborhood structures in sense of Lowen, are special global L-neighborhood groups, for L=I is the closed unit interval.

Completion of L-topological groups

The goal of this paper is to extend an L-topological group to a complete L-topological group, which necessitates formalizing the completion of an L-topological group. In so doing, we introduce the notion of the completion of an L-uniform space in the sense of Gahler, Bayoumi, Kandil and Nouh.

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 a -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 Gahler et al. (1998) which is defined as some fuzzy filter and we are also interested in the fuzzy proximity N in the sense of Gahler 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 Uα and the α-level proximity Nα U and N, respectively. Weshow that there is one-to-one correspondence between a fuzzyuniform structure U and the family (Uα)α∈L0 of uniform structuresthat fulfills certain conditions, is given by: Uα=Uα and U(U)=⋁A∈Uα,χA⩽uα. We also show that the topologies Tuα and TNα associated with Uα and Nα coincides with theα-level topologies of the fuzzy topologies τUand τN associated to U and N,respectively, that is, TUα=(τU)α and TNα=(τN)α. Moreover, we assign for each fuzzyuniform structure U an associated fuzzy proximity of theinternal type NU and hence we get the relationbetween the α-levels of U and of NU which is given by: NUα=(NU)α.

On L-proximities of the internal type

In this paper, the initial and final L-proximities, for a notion of L-proximity introduced by the author and others in 1998, are investigated. This L-proximity is called L-proximity of the internal type. This paper shows that all initial and final lifts in the category L-PRI of L-proximity spaces of the internal type and hence all initial and final L-proximities of the internal type exist. The expected relation between the initial L-proximities of the internal type and the initial L-topologies, the initial global L-neighborhood structures, the initial L-uniform structures is verified. That is, the L-topology (global L-neighborhood structure) associated with the initial of a family of L-proximities of the internal type coincides with the initial of the family of L-topologies (global L-neighborhood structures) associated with these L-proximities of the internal type. Moreover, the L-proximity of the internal type associated with the initial of a family of L-uniform structures coincides with the initial of the family of L-proximities of the internal type associated with these L-uniform structures.