Hierarchies and Uncertainty Measures on Pythagorean Fuzzy Approximation Spaces

Abstract

In Granular Computing, the hierarchies and uncertainty measures are two important concepts to investigate the granular structures and uncertainty of approximation spaces. In this paper, hierarchies and uncertainty measures on pythagorean fuzzy approximation spaces will be researched. Firstly, the introduction and operations of pythagorean fuzzy granular structures are given, and three hierarchies and a lattice structure on pythagorean fuzzy approximation spaces are examined. The hierarchies are characterized by three order relations, the first order relation is defined on the inclusion relation of pythagorean fuzzy information granules, the second one is defined on the cardinality of pythagorean fuzzy information granules, and the third one is defined on the sum of the cardinality of pythagorean fuzzy information granules. The lattice structure is constructed on the first order relation on pythagorean fuzzy approximation spaces. Fuzzy information granularity and fuzzy information entropy are extended to describe the uncertainty of pythagorean fuzzy granular structures, and the relationship between the uncertainty measures and hierarchies are discussed. The examples show that hierarchies are effective to analyze the relationships among all granular structures on pythagorean fuzzy approximation spaces.