Some Issues About GV-like Fuzzy Matroids

GV-like fuzzy matroids (called also fuzzy motriods on the 3-element lattice) have been defined and studied based on the ide of specialization. In this thesis, we continue to study some issues left over. Firstly, introduce some concepts and conclusions on matroid theory and fuzzy mathematics which will be used in this paper. Secondly, introduce families of independent sets of GV-like fuzzy matroids, proves that GV-like fuzzy matroid and perfect L-matroid are the same notion (where L={0,1/2,1} called also three-element lattice), and that a GV-like fuzzy matroid can be determined by a tower of matroids. Thirdly, introduce families of bases of GV-like fuzzy matroids, defines the notion of family of bases of GV-like fuzzy matroids, and proves that a family of independent sets of a GV-like fuzzy matroids can be determined by a tower of families of matroid bases, fourthly, define rank functions of GV-like fuzzy matroids, is an improvement or a modification to [15], in which the notion of rank function of a GV-like matroid is redefined, and two mappings Ψ_S:Ⅱ(S, L) →R(S, L) and Ψ_S:R(S, L) → Ⅱ(S, L) satisfying Ψ_S~ΟΨ_S=idⅡ(S,L) are defined, where Ⅱ(S, L) is the set of all families of independent sets of GV-like fuzzy matroids, and R(S, L) is the set of all rank functions of GV-like fuzzy matroids. At last, build two kinds of modular lattices related to the three-element lattice. Proves that both the set of all sub-L-linear spaces of an L-linear space and the set of all normal sub L-subgroups of an L-subgroups are modular lattices.