The α-embedded and A's fussy decomposition theorems

Fuzzy set theory becomes more popular in recent years due to its simplicity in modeling and effectiveness in control problems. Unfortunately, both fuzzy union-decomposition and fuzzy intersection-decomposition problems for fuzzy set A did not attract too much attention by the fuzzy researchers and thus not many people studied on these topics during the past 30 years. This paper gives the concepts of α-embedded set A^α for fuzzy set A. On this basis, we propose the fuzzy union-decomposition theorems for fuzzy set A, the fuzzy intersection-decomposition theorems for fuzzy set A, the relational theorems of fuzzy union-decomposition and fuzzy intersection-decomposition for fuzzy set A.