Supermatroids And Fuzzy Matroids

Matroids, as an abstract theory of dependence of graphs and linear algebra, was firstintroduced by Whitney in1935. Matroid theory now becomes an important branch ofcombinatorial mathematics. Generalizaitons of matroids have been an important aspect inmatroid study. This thesis concentrates on the generalizations of matroids.Matroids are combinatorial structures defined on finite sets. According to wether thestructured-sets are finite sets (posets), we divide the existed generalizations of matroids intotwo cases: finite generalizations and infinite generalizations. As for finite cases, supermatroidsand greedoids are two import generalizations. This thesis studies mainly axioms ofsupermatroids on modular lattices. And for infinite generalizations, we do study the relationsof existed fuzzy matroids and connectedness of a special kind of fuzzy matorids.1. Based on supermatroids, this thesis gives a survey of finite generalizations of matroids.2. To generalize the axiom systems of DL-matroids (distributive supermatroids), baesaxiom and independent set axiom are set up for ML-matroids. Examples are given to showthat related results are not correct for supermatroids on submodular lattices. Circuits insupermatroids are studied and circuit elimination of DL-matroids is proved directly. Based onthese results concerning circuits in DL-matroids, we may construct the thanstivity of circuits,and then study the connectedness of DL-matroids in future.3. Connectedness of fuzzy matroids is first introduced. Thanstivity of fuzzy circuits isproved for G-V fuzzy matroids under some conditions. A new class of G-V fuzzy matroids,refined G-V fuzzy matroids, are defined and their properties are studied. Refined G-V fuzzymatroids are regular and Thanstivity of fuzzy circuits hold. Connectedness of refined G-Vfuzzy matroids is studied.4. Five kind of fuzzy matroids are compared.5. A new class of fuzzy matroids are induced by polymatoids, and this kind of fuzzymatroids are connected firmly with H fuzzy matroids. Following the idea of polymatroids, aclass of H matroids are defined by considering a fuzzy set on a three-point set as a point inEuclidean space. This special class of H matroids give an example to show that r-levels of Hmatroids need not be matroids, which solves an open problem about the relation between Hfuzzy matroids and fuzzy pre-matroids. In addition, we characterize H fuzzy matroids ontwo-point sets graphically.