In this paper, we introduce the notions of L-fuzzy quasi-uniformities and Lfuzzy interior operators in complete residuated lattices. We investigate the L-fuzzy quasiuniformities induced by L-fuzzy interior operators. We study the relationships between Lfuzzy interior operators and L-fuzzy quasi-uniformities. We give their examples. AMS Subject Classification: 03E72, 06A15, 06F07, 54F05

This chapter introduces a rule-based perspective on the framework of fuzzy lattices, and the Fuzzy Lattice Reasoning (FLR) classifier. The notion of fuzzy lattice rules is introduced, and a training algorithm for inducing a fuzzy lattice rule engine from data is specified. The role of positive valuation functions for specifying fuzzy lattices is underlined and non-linear (sigmoid) positive valu...

Hierarchy plays a crucial role in the development of the granular computing(GrC). Presently, a lot of hierarchies have been studied for characterizing the finer or coarser relationships among information granulations, which are derived from the crisp binary relations. To further improve the existing results, not only the fuzzy information granulations derived from the fuzzy binary relations are...

This paper gives analogies of Hutton’s quasi-uniformities for complete lattices and extensions of Shi’s pointwise quasi-uniformities to closed set lattices.CLGOH denotes the category of all complete lattices with generalized order homomorphisms asmorphisms and CSL the category of all closed set lattices.HQUnif and SPQUnif denote the categories of Hutton quasi-uniform spaces on complete lattices...

Starting with the underlying motivation of developing a general theory of L-fuzzy sets where L is a multilattice (a particular case of non-deterministic algebra), we study the relationship between the crisp notions of congruence, homomorphism and substructure on some non-deterministic algebras which have been used in the literature, i.e. hypergroups, and join spaces. Moreover, we provide suitab...

A group Γ is defined to be cofinitely Hopfian if every homomorphism Γ → Γ whose image is of finite index is an automorphism. Geometrically significant groups enjoying this property include certain relatively hyperbolic groups and many lattices. A knot group is cofinitely Hopfian if and only if the knot is not a torus knot. A free-by-cyclic group is cofinitely Hopfian if and only if it has trivi...

Vagueness and high dimensional space data are usual features of current data. The paper is an approach to identify conceptual structures among fuzzy three dimensional data sets in order to get conceptual hierarchy. We propose a fuzzy extension of the Galois connections that allows to demonstrate an isomorphism theorem between fuzzy sets closures which is the basis for generating lattices ordere...

We focus on some important fuzzy logic algebras on residuated lattices. These interesting algebraic structures and their main properties are reviewed, and their relationships are clarified. Corresponding formal deductive systems for fuzzy logic are