Şenol Dost

Ditopological texture spaces and fuzzy topology, I. Basic concepts

This is the first of three papers which develop various fundamental aspects of the theory of ditopological texture spaces in a categorical setting and present important links with the theory of L-topological spaces. The authors begin by defining the notion of q-sets, which together with the p-sets considered earlier, enable the formulation of a powerful concept of duality. This plays an important role in the theory of direlations and difunctions, which is described here in detail. Difunctions are then taken as the morphisms of a category dfTex, whose objects are texture spaces. Several important subcategories are defined and the closely related construct fTex defined. Some properties of the functors between these categories are obtained.

Ditopological texture spaces and fuzzy topology, II. Topological considerations

This is the second of three papers which develop various fundamental aspects of the theory of ditopological texture spaces in a categorical setting and present important links with the theory of L-topological spaces. In the first paper in this series, subtitled Basic Concepts, the authors presented the notions of direlation and difunction between textures and introduced the category dfTex, the construct fTex and several related categories. In this paper the category dfDitop of ditopological texture spaces and bicontinuous difunctions is defined and the forgetful functor U:dfDitop→dfTex shown to be topological. Several properties are discussed, including the existence of products and coproducts. An equivalence with the category of classical Hutton spaces is presented and the paper ends with a consideration of L-valued sets and topologies from the viewpoint of ditopological texture spaces.

Interior and closure operators on texture spaces---I: Basic concepts and C̆ech closure operators

This paper is the first of a series of three papers on the theory of interior and closure operators. Here, the theory is discussed from the textural point of view. First, the interior and closure operators on texture spaces are defined and some basic properties are given in terms of neighbourhoods and coneigbourhoods. Then the category dfIC whose objects are interior-closure spaces and the morphisms are bicontinuous difunctions is shown to be topological over the ground category dfTex of textures and difunctions. Further, considering the closure operator on Hutton algebras (known as fuzzy lattices) in the sense of C@?ech, the category HutCl of Hutton closure spaces and continuous mappings is defined. Finally, the category cdfIC of complemented bicontinuous difunctions and complemented interior-closure texture spaces and the opposite category of HutCl are shown to be equivalent.

Interior and closure operators on texture spaces---II: Dikranjan--Giuli closure operators and Hutton algebras

In this work, we discuss interior and closure operators on textures in the sense of Dikranjan-Giuli. First, we define the category dfICL of interior-closure spaces and bicontinuous difunctions and show that it is topological over dfTex whose objects are textures and morphisms are difunctions. The category L-CLOSURE of L-closure spaces and Zadeh type powerset operators, and the counterparts of the Lowen functors have been presented by Wu-Neng Zhou in a fixed-basis setting. We consider the closure operators on a Hutton algebra L and, in a natural way, we define the category HCL of Hutton closure spaces taking the morphisms of the opposite category of HutAlg-the category of Hutton algebras (fuzzy lattices) and the mappings preserving arbitrary meets, joins and involution. In this case, the categories L-CLOSURE and H-the category of Hutton spaces and the morphisms in the sense of Definition 2.1-can be considered as a subcategory and a full subcategory of HCL, respectively. Using the fact that dfICL and HCL^o^p are equivalent categories, we guarantee the existence of products and sums in HCL. Finally, we show that the generalized Lowen functor can be also given in a textural framework for [0,1].

A textural view of semi-separation axioms in soft fuzzy topological spaces

In the present paper, we have continued to study the properties of semi-separation axioms in ditopological texture spaces. We prove that the semi-Ti-spaces (i = 0, 1, 2, 3, 3 1 2 , 4) are productive in ditopological spaces. Finally, semi-separation axioms are given for soft fuzzy topological spaces by using the categorical isomorphism between soft fuzzy topologies and ditopologies.

A textural view of soft fuzzy rough sets

This paper aims to give a new insight for soft fuzzy sets. Firstly, using textural soft fuzzy direlations on a soft fuzzy lattice, we obtain a soft fuzzy rough set algebra where the inverse soft fuzzy relation and inverse soft fuzzy corelation are the upper approximation and lower approximation, respectively. On the other hand, we consider Alexandroff soft fuzzy topology and we prove that there exists a one-to-one correspondence between the Alexandroff topologies and the reflexive and transitive direlations on a texture space.