The Study On The Fuzzy Transversal Matroids

This paper mainly studies the related problems of the fuzzy transversal matroids,enriching and expanding the theory of transversal and transversal matroids,fuzzy transversal and fuzzy transversal matroids gradually.The innovations in this article are as follows:1.The paper has studied the essential relationship between transversal and fuzzy traversal.That is,a necessary and sufficient condition for a fuzzy set to be a fuzzy traversal on a family of fuzzy sets is that the support set of the fuzzy set is the traversal on the corresponding family of subsets,and satisfies the corresponding fuzzy membership degree limit.The paper has defined the fuzzy bipartite graph,and described the fuzzy partial transversal by the fuzzy matching property of the fuzzy bipartite graph.That is,a necessary and sufficient condition for a fuzzy set to be a fuzzy partial traversal on a family of fuzzy sets is that the fuzzy set is covered by a fuzzy matching of the fuzzy bipartite graph.2.With help of a counterexample,the paper has proposed the suspicions of the definition of the fuzzy transversal matroids.It is pointed out that not all fuzzy partial transversals on some families of fuzzy sets can form fuzzy matroids.According to the essential definition of fuzzy partial transversal and the quondam definition fuzzy transversal matroid,the paper has analyzed the reason of happening the problem.That is,fuzzy partial transversal has an element of fuzzy membership limit,but the partial transversal doesn’t have the element.Many methods have been proposed to solve this problem,such as changing the definition of the fuzzy set,changing the definition of the fuzzy partial traversal,changing the definition of the fuzzy matroid or changing the definition of the fuzzy transversal matroids.3.The paper has redefined the fuzzy transversal matroids,and has found one class of fuzzy set families which all fuzzy partial transversals on a fuzzy sets family of the class can form a fuzzy matroid,called as the uniform fuzzy transversal matroids.The paper has also studied many properties about fundamental sequences,induced matroid sequences and fuzzy bases in uniform fuzzy transversal matroids and analyzed the relation between uniform fuzzy transversal matroids and quasi-fuzzy graph matroids.It is proved that uniform fuzzy transversal matroids are one class of quasi-fuzzy graph matroids.These studies and conclusions have pointed out the problems on the research of fuzzy partial transversals by the previous scholars,then have proposed and deeply researched a new definition of the fuzzy transversal matroid,called as the uniform fuzzy transversal matroids.These studies have enriched the theory of the fuzzy transversal matroids.