| In this paper, seven topics for the structure of the fuzzy independence set system and fuzzy matroid are investigated based on the existed theories, stated as follows :1) The properties of fuzzy pre-independence set system are studied in detail, and then, several necessary and sufficient conditions for the family of level sets of a closed fuzzy pre-independence set system are derived.2) Several properties of fuzzy independence set system are presented. Moreover, based on the study on the properties of the fuzzy pre-independence set system, the existence of the base of fuzzy independence set system is also investigated.3) The concept of fuzzy pre-matroid is first proposed. Then its properties are analyzed. Moreover, two necessary and sufficient conditions for the existence of fuzzy pre-matroid are given. 4) For fuzzy independence set system and fuzzy independence space, the dissimilarity of properties of their bases is analyzed. 5) Introducing the idea of crisp matroid sums to fuzzy matroid, I address the definition of fuzzy matroid sum, which is studied in details in terms of its properties. Particularly, several results for sum of close and regular fuzzy matroids are obtained. These are obviously important to study the structure of fuzzy matroids.6) By generalizing the idea of crisp matroid products to fuzzy matroid, the fuzzy matroid product is defined. The properties of fuzzy matroid products are studied in detail. Several important results for the base and existence of these products are obtained.7) Based on the study on the properties of the fuzzy relationship, several properties of fuzzy closure operators are presented. All of these seven aspects of fuzzy independence set system and fuzzy matroids are the important contents of the fuzzy independence set system and fuzzy matroids, and the research on them is the base for further study. |
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