Rıza Ertürk

Fuzzy sets as texture spaces: I. Representation theorems

The authors continue the development of a theory of texture spaces, introducing complemented products and sums, and applying these in a series of representation theorems for fuzzy lattices and the lattices of L-fuzzy sets, generalized fuzzy sets and intuitionistic sets. The second paper in this series will extend this enquiry by introducing subtextures and quotient textures and a second series of papers is under preparation which consider topological aspects of this correspondence.

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.

Fuzzy sets as textures spaces, II. Subtextures and quotient textures

In this paper the authors introduce the notions of subtexture and quotient texture of a complemented texture space. Conditions ensuring the existence of such textures are discussed, and applied to show that certain natural subtextures of product textures may be defined. These are used in particular to give representations of product L-fuzzy sets, and a representation involving quotient textures is also examined.

Ditopological texture spaces and fuzzy topology—III: Separation axioms

This is the third of three papers which develop various fundamental aspects of the theory of ditopological texture spaces and relate this work with the theory of fuzzy topological spaces. In this paper, the authors define and study basic separation axioms in general ditopological texture spaces, relating these to known separation axioms for bitopological and topological spaces, and to the point-free separation axioms of B. Hutton and I. Reilly for fuzzy topological spaces.

Separation axioms in fuzzy topology characterized by bitopologies

Abstract In this paper we associate a bitopological space with a fuzzy topological space, and examine the relation between various fuzzy and bitopological separation axioms.

Some results on fuzzy compact spaces

Abstract In this paper we examine first the relation between the fuzzy compactness of a fuzzy topology on a fuzzy lattice L and the compactness of two topological spaces ( M , u ), ( M , v ) defined on the set M of molecules of L . We then investigate the relation between fuzzy compactness and the joint compactness of the bitopological space ( M , u , v ).