Dexue Zhang

Fuzzy preorder and fuzzy topology

This paper presents, by means of Alexandrov topology for fuzzy preordered sets and specialization order for fuzzy topological spaces, a systematic investigation of the interrelationship between fuzzy preordered sets, topological spaces, and fuzzy topological spaces.

Concept lattices of fuzzy contexts: Formal concept analysis vs. rough set theory

This paper presents a comparative study of concept lattices of fuzzy contexts based on formal concept analysis and rough set theory. It is known that every complete fuzzy lattice can be represented as the concept lattice of a fuzzy context based on formal concept analysis [R. Belohlavek, Concept lattices and order in fuzzy logic, Ann. Pure Appl. Logic 128 (2004) 277-298]. This paper shows that every complete fuzzy lattice can be represented as the concept lattice of a fuzzy context based on rough set theory if and only if the residuated lattice (L,*,1) satisfies the law of double negation. Thus, the expressive power of concept lattices based on rough set theory is weaker than that of concept lattices based on formal concept analysis.

An enriched category approach to many valued topology

In this paper we discuss a basic question in fuzzy set theory: what is a many valued topology on a set X? An answer is presented by interpreting the axioms (two logical formulas) for classical topological spaces in the fuzzy setting making use of the intrinsic fuzzy partial order on the fuzzy power set of X. As preparation, a systematic investigation of fuzzy partial orders is presented from the viewpoint of enriched category theory.

Categories isomorphic to FNS

The main result in this paper is that the category of fuzzy neighborhood spaces is isomorphic to that of fuzzifying topological spaces.

Triangular norms on partially ordered sets

In this paper, several methods to construct triangular norms on partially ordered sets are proposed. These methods include:(1)Constructing triangular norms on the poset of closed intervals of a given bounded poset. (2)Constructing triangular norms via Galois connections between bounded posets based on a categorical analysis of the construction of triangular norms on closed intervals via monotone functions and their pseudo-inverses. (3)Constructing triangular norms on function spaces between partially ordered sets.

Fuzzy Galois connections categorically

This paper presents a systematic investigation of fuzzy (non-commutative) Galois connections in the sense of R. Bělohlavek [1], from the point of view of enriched category theory. The results obtained show that the theory of enriched categories makes it possible to present the theory of fuzzy Galois connections in a succinct way; and more importantly, it provides a useful method to express and to study the link and the difference between the commutative and the non-commutative worlds (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The concept lattice functors

This paper is concerned with the relationship between contexts, closure spaces, and complete lattices. It is shown that, for a unital quantale L, both formal concept lattices and property oriented concept lattices are functorial from the category L-Ctx of L-contexts and infomorphisms to the category L-Sup of complete L-lattices and suprema-preserving maps. Moreover, the formal concept lattice functor can be written as the composition of a right adjoint functor from L-Ctx to the category L-Cls of L-closure spaces and continuous functions and a left adjoint functor from L-Cls to L-Sup.

L-Fuzzifying topologies as L-topologies

It is proved in this paper that for every completely distributive lattice L with a locally multiplicative way below relation the category of L-fuzzifying topological spaces can be embedded in the category of L-topological spaces (stratified Chang-Goguen topological spaces) as a simultaneously bireflective and bicoreflective full subcategory. Several equivalent descriptions of this subcategory are given.

Preordered sets valued in a GL-monoid

Let M=(L,*) be a GL-monoid. An M-valued preordered set is an L-subset endowed with a reflexive and M-transitive L-relation, it is essentially a category enriched in a quantaloid generated by M. This paper presents a study of M-valued preordered sets with emphasis on symmetrization and the Cauchy completion. The main result states that symmetrization and the Cauchy completion of M-valued preordered sets commute up to a natural isomorphism.

Stratified Hutton uniform spaces

In this paper some categorical properties of Hutton uniform spaces are investigated. At first, the initial and final structures for Hutton uniform spaces are discussed and it is proved that the canonical functor from the construct of Hutton uniform spaces to that of L-topological spaces preserves initial sources. Secondly, the notion of stratified Hutton uniform spaces is introduced, and it is demonstrated that every stratified Hutton uniformity induces a stratified L-topology and that the relationship between stratified Hutton uniform spaces and Hutton uniform spaces is like that between stratified L-topological spaces and L-topological spaces. Finally, it is showed that the construct of uniform spaces can be embedded in the construct of stratified Hutton uniform spaces as a concretely both reflective and co-reflective full sub-construct.