The structural analysis of fuzzy measures
From quotient space based granular computing theory we explore the fuzzy measures of fuzzy sets. Based on the structural analysis of fuzzy sets, we present an "isotropism" assumption. Under this assumption we achieved the following insights: (1) the uniqueness of a fuzzy measure being isotropic on a finite complete semi-ordered set; (2) the necessary and sufficient condition of the isomorphism for fuzzy measure functions in fuzzy mathematics; (3) the necessary and sufficient condition that fuzzy measures have fuzzy and granular monotonicity; (4) under a certain assumption, the analytic expression for fuzzy measures. The results clarify the relationship between fuzzy measures and granular computing, reveal the essence of fuzzy measures, and provide a simple way of constructing fuzzy measures.
structural analysis fuzzy measure quotient space fuzzy set rough set