Lawrence M. Brown

Ditopological texture spaces and intuitionistic sets

Abstract In this paper it is shown that the lattice of intuitionistic subsets of a set X in the sense of D. Coker may be represented as a special type of texture space, called an intuitionistic texture on X , and various characterizations are given. It is established that intuitionistic topologies are mapped to ditopologies on the corresponding texture, and some notions of compactness and stability are considered.

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.

Categories of dicompact BI- T 2 texture spaces and a Banach-stone theorem

The principal aim of this paper is to consider various aspects of the theory of dicompact bi-T 2 texture spaces, and place them in a categorical setting. It culminates in a version of the Banach-Stone theorem. On the way, a new class of textures, called here nearly plain textures, is seen to play a crucial role in the development of the theory.

Extended real dicompactness and an application to Hutton spaces

Abstract The authors extend the notion of real dicompactness from the class of nearly plain textures to the considerably larger class of almost plain textures and show how this may be used to define a notion of realcompactness for Hutton spaces.

Semi-sequentially normal bitopological spaces

Abstract In this paper we introduce a weak form of full normality for bitopological spaces, and consider its relationship to pairwise paracompactness in the sense of S. Romaguerra and J. Marin, and to the notion of σ-bicushioned refinement for dual covers.

Di-uniformities and Hutton uniformities

The authors characterize di-uniformities on a texture (S,S) in the sense of Ozcag and Brown (Di-uniform texture spaces, Appl. Gen. Top. 4(1) (2003), 157-192) in terms of functions on the texturing S. This characterization enables quasi-uniformities in the sense of Hutton (Uniformities on Fuzzy Topological Spaces, J. Math. Anal. Appl. 58 (1977) 559-571) to be regarded as di-uniformities on the corresponding Hutton Texture, thereby revealing di-uniformities as a generalization of Hutton quasi-uniformities. The effect of imposing a complementation on (S,S) is also considered and several important results established.