Hierarchies on fuzzy information granulations: A knowledge distance based lattice approach

Abstract

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 presented in this paper, but also the three orderings are explored for expressing the hierarchies on fuzzy information granulations. Furthermore, we also introduce the concepts of the global and local knowledge distances into fuzzy information granulations, which can be used to construct the knowledge distance lattices. It shows that the derived partial orderings through those lattices are corresponding to the three different hierarchies on fuzzy information granulations, respectively. The theoretical results provide us a more comprehensible perspective for the study of granular computing(GrC).

Hierarchies and Uncertainty Measures on Pythagorean Fuzzy Approximation Spaces

“…where SP(x i ) ⊆ SQ(x i )(1 ≤ i ≤ n) means that each membership of the elements in SP is less than or equal to that in SQ, that is, p i j ≤ q i j (1 ≤ i ≤ n) for each element x i . Definition 4 [11] Let F(P), F(Q) ∈ F(U,Ω), F(P) = (SP(x 1 ),…”

“…In order to explore the finer or coarser relationships among multigranulation spaces, Yang et al [10] have conducted three different hierarchies on the multigranulation rough set models. Song et al [11] introduced knowledge distance [12,13] to fuzzy environment, and proposed the local knowledge distance and global knowledge distance, such two knowledge distances are used to construct algebraic lattices, which are useful in characterizing the hierarchies on fuzzy information granulations. Huang et al [14] investigated the hierarchies and a lattice structure of intuitionistic fuzzy set, and then further researched the uncertainty measures of intuitionistic fuzzy set.…”