The Properties Of Matroid Operator,Matroid Mapping And Matroid Category

Abstract: Matroid theory is not an 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. After that, matroid theory had further development, but the main ideas of studying matroid come from graph theory and lattice. In this paper, the idea of totopology penetrates the study of matroid theory. Some concepts are introduced such as continuous mapping, open mapping,closed mapping, homeomorphic mapping, submatroids and quotient matroids. and many kinds of opetators. and some equivalent conditions of morphisms in matroid category are studied .The construction and the main contents of this paper are as follows:In chapter 1 We maked a depiction for basic knowledge and basic conclusions of matroid ,topology and category which would be used in the following chapters.In chapter 2 We give out the defitions of closure operators, interior operators, exterior operators , completion operators and boundary operators and prove that we can ensure a unique matroid by one of the closure operator, interior operator, exterior operator or boundary operator in a finite set. In addition, the Kuratowski Fourteen sets Theorem in matroid theory is proved. At last the properties of new operators compounded by these five are studied.In chapter 3 The properties of matroid mappings and matroid category are studied. At first, we give out the defitions of continuous mapping, open map-ping,closed mapping ,homeomorphic mapping,and isomorphism in the matroids, and we obtain some equivalent conditions. And then,we proved that homeomorphic in the matroid and isomorphism that defitied by us in this paper are equivalent;We studied the problem whether the open sets, closed sets, independent sets, dependent sets and circuit were keeped under the continuous mapping.Second, the submatroids and quotient matroids are defined. The relations...