Some Researches On Fuzzy Matroids

The concept of fuzzy matroid was introduced by Goetschel and Voxman in 1988. Integrating with different subjects is a typical property in the research of matroids, which makes the matroids theory rich and strong. The theory of fuzzy matroids is such the successful integration of the matroids theory and the fuzzy sets theory, which is involved in the theory of fuzzy graph, partial sets and lattice, and combinatorial mathematics. In this paper, we firstly study the axiom systems of fuzzy matroids and their properties. Until now, apart from the independent axiom, there is no other axiom system which is equivatent to the independent axiom in the theory of fuzzy matroids as that in the matroid theory. So the study of the axiom systems of fuzzy matroids will improve and complement the theory of fuzzy matroids. Using the relation of fuzzy dependence, we obtain the closure axioms and the closed sets axioms of of elementary fuzzy matroids. By studying the section fuzzy matroids, we get the properties of the bases of fuzzy matroids, and give the equivalent characteristics of the bases of fuzzy matroids, then we obtain the base axiom of fuzzy matroids. These work lay a solid foundation on studying the theory of fuzzy matroids. Secondly, we define the fuzzy transversal and the partial fuzzy transversal of a finite family of fuzzy sets. Because the first and most basic problem in the transversal theory is finding conditions for a family of sets to have a transversal, we investigate the conditions for a family of fuzzy sets to have a fuzzy transversal and a partial fuzzy transversal. We proved that the collection of partial fuzzy transversal of a family of fuzzy sets formed the set of fuzzy independent sets of a fuzzy matroid, and we also discussed the fuzzy Hall theorem and fuzzy Rado theorem of a family of fuzzy sets that their underlying set is an infinite set. Now we give the main content of this paper as follows:1. Two axiom systems of elementary fuzzy matroids and some properties of fuzzy matroids are studied. Firstly, several operations of fuzzy matroids are discussed, including two k-truncations, restriction and contraction of a fuzzy matroid under a fuzzy set. The sum of fuzzy matroids are also investigated. These operations not only observe the properties in the matroid theory, but also have new characteristics in fuzzy matroids. Secondly, the concept of fuzzy dependence and fuzzy closure operator are introduced, and the properties of fuzzy dependence and fuzzy closure operator are investigated, then the closure axioms and the closed sets axioms of of elementary fuzzy matroids are obtained. Finally, by studying the properties of fuzzy circuit, a new connectivity of fuzzy matroids is defined, and a series of the properties of this connectivity are studied.2. The base axioms of fuzzy matroids and some properties of fuzzy bases are studied systematically. Firstly, the section fuzzy matroids are defined and the properties of their bases are investigated, then we get the base axioms of the section fuzzy matroids. Secondly, we study the properties of the bases of fuzzy matroids, then the relationships of the bases between the fuzzy matroids and the section fuzzy matroids are discussed. Some equivalent characteristics of the bases of fuzzy matroids are given. In the end, we obtain the base axioms of the closed regular fuzzy matroids.3. The fuzzy transversal theory is studied and some properties are investigated. Firstly, the concept of the fuzzy transversal and the partial fuzzy transversal are introduced, then the fuzzy Hall theorem and fuzzy Rado theorem are investigated, respectively. We prove that the collection of partial fuzzy transversal of a family of fuzzy sets formed the set of fuzzy independent sets of a fuzzy matroid. Secondly, the p-fuzzy transversal of a family of fuzzy sets is defined and some properties are investigated. We also get some necessary and sufficient conditions that a family of fuzzy sets have a p-fuzzy transversal and this p-fuzzy transversal is the set of fuzzy independent sets. Finally, we study the fuzzy transversal theory on an infinite set. By the fuzzy Rado's selection principle, we study the fuzzy Hall theorem and fuzzy Rado theorem of the fuzzy sets whose supporting set is an infinite set.