Properties Of Fuzzy Matroids On Three-element Lattices

Matroid theory is not an old branch with a long history in comparison with other mathematical branches. In 1935, Whitney analyzed abstract properties of linear dependence of vectors, and established the concepts of matroids. In 1988, R. Goetschel and W. Voxman introduced " fuzzy " concepts into matroid theory, and then started the research of fuzzy matroids. In this thesis, we introduced concepts of L-fuzzy sets and L-fuzzy matroids, and studied main properties and characteristics of L-fuzzy matroids, where L is the simplest nontrivial fuzzy set.The construction and the main contents of this thesis are as follows.In chapter 1, we maked a depiction for basic knowledge and basic conclusions of fuzzy mathematics and matroids which would be used in the foUowing chapters.In chapter 2, the concept of L-fuzzy matroids was given, and then some new matroids and new .L-fuzzy matroids were constructed from the given L-fuzzy matroids, such as cut matroids, direct sum of L-fuzzy matroids, restriction of L-fuzzy matroids, uniform L-fuzzy matroids, elementary L-fuzzy matroids, etc. Then we studied the relations between L-fuzzy matroids and their cut matroids, and the equivalent condition for a fuzzy set to be an independent set. Some good conclusions were obtained, which paved the way for research of the following chapters.In chapter 3, the fuzzy bases and fuzzy circuits of L-fuzzy matroids were mainly studied.Firstly, the definition of fuzzy bases, and the necessary and sufficient conditions for a fuzzy set to be a fuzzy base, and the relations between fuzzy bases and the bases of the cut matroids were given. Secondly, we defined the concepts of closed L-fuzzy matroids and regular L-fuzzy matroids, and then got several main conclusions, as follows:(1) The support sets of bases of an L-fuzzy matroid have the same cardinality.(2) In regular If-fuzzy matroids, the necessary and sufficient condition for a fuzzy set to be a fuzzy base is that the cut set of the fuzzy set is the base of the corresponding cut matroid.(3)A necessary and sufficient condition for a L-fuzzy matroid to be regular is that the bases of L-fuzzy matroid have the same cardinality.