The Discussion About The Necessary And Sufficient Conditions Of Fuzzy Matroids

Matroids have many equivalent descriptions(which are also called necessary and sufficient conditions)by means of the concepts of "base","circuit","rank","closure" and "hyperplane".The study of these equivalent descriptions of matroids is important to matroid theory.The concept of fuzzy matroid can be generalized by the concept of matroid,when ordinary sets in matroids are extended to fuzzy sets.Therefore,with the help of the research methods of matroid theory,people have discussed the necessary and sufficient conditions of fuzzy matroids.However,the complexity of fuzzy matroids makes it difficult to study the equivalent descriptions of fuzzy matroids.This paper firstly summarizes necessary and sufficient conditions of various fuzzy matroids.Then,many properties of fuzzy circuits in fuzzy matroids are analyzed.Finally,from these properties of fuzzy circuits,a necessary and sufficient condition of closed fuzzy matroids is studied.The main works in the paper are as following:(1)The necessary and sufficient conditions of all kinds of fuzzy matroids are summarized,and their characteristics,methods of proofs,scope of use,and interconnections between them are analyzed.(2)Properties of the circuits of derive matroids are analyzed,and some important properties of the fuzzy circuits of closed fuzzy matroids are obtained,which is necessary for studying the necessary and sufficient conditions of fuzzy matroids through elementary fuzzy circuits set later.(3)Circuit function and circuit interval are generalized to obtain generalized circuit function and circuit range,which further enrich the properties of fuzzy circuits in closed fuzzy matroids.(4)The properties of elementary fuzzy circuits are in-depth analyzed,and five main properties including normalization,anti-inclusion,existence,generalized circuit function and synthesis are drawn.(5)A necessary and sufficient condition for closed fuzzy matroids(which is called fuzzy circuit axiom of closed fuzzy matroids)is obtained in terms of above properties of elementary fuzzy circuits set.And a new method for studying the necessary and sufficient conditions of fuzzy matroids is provided.The research on fuzzy circuits,elementary fuzzy circuits and circuits of derived matroids has enriched fuzzy matroid theory.This research is beneficial to the understanding of the relationship between matroids and fuzzy matroids.In particular,fuzzy circuit axiom of closed fuzzy matroids profoundly reveal the intrinsic nature of closed fuzzy matroids,and describe closed fuzzy matroids from a completely new perspective.These works are significant to further study fuzzy matroids.