Abstract: The three-way concept analysis is a combination of three-way decision and formal concept analysis. The greatest progress of this theory compared with formal concept analysis is that it can simultaneously study the information that is “commonly shared” and “not commonly shared” in the formal context. Attribute granu-lation is a theory based on the decomposition of attributes into sub-attributes using a granularity tree and pruning, forming a new set of attributes. However, due to the numerous prunings on the same granularity tree, the key issue to ensure the efficiency of attribute granulation is how to choose the pruning and determine the optimal direction for further operations to achieve optimal granulation results. In this paper, through theoretical derivations, it is proved that there is a close internal relationship between the original three-way concepts and the new three-way concepts obtained from attribute granulation, which can be used as the basis for measuring the efficiency of attribute granulation. Firstly, based on the relationship of attribute granulation levels, the attribute granulation levels are divided into attribute granulation levels with partial order relationships and attribute granulation levels without partial order relationships. Furthermore, the definition of refinement coefficients is given, and the measurement roles of refinement coefficients in the two types of attribute granulation levels are respectively explained, so as to achieve the purpose of measuring the efficiency of different attribute granulation directions.

Key words: three-way concept, granularity of attributes, multi-granularity, coefficient of elaboration, granularity tree