Constructive Operations In Extended Theories Of Matroids

Matroid theory is an important part of combinatorial mathematics and discrete mathematics. It has been generalized and some new theories based, such as poset matroids, greedoids and fuzzy matroids. Because of the one to one correspondence between finite posets and finite distributive lattices, we can give an euivalent defination of poset matroids that uses the language of distributive lattices and call it combinatorial scheme. In this article, the extended theories of matroids are poset matroids (combinatorial scheme) and fuzzy matroids. We can construct new matroids by basic matroid opetations, such as truncation, expansion, restrication and contraction, etc. We will study these operations in poset matroids (combinatorial scheme) and fuzzy matroids. The main content of this paper is as follows:1. This paper gives a necessary and sufficient condition ensuring that the restriction and contraction to same subset of a poset matroid processes the same result, which indicates that the rank function plays an important role in the study of poset matroids. And then we discuss some operations on combinatorial schemes, such as restriction, contraction, truncation and elongation. Some properties of these operations are also studied. This paper gives definitions of the closure operator, interior operator and complement operator in poset matroids, of which the underlying posets have an order-reversing involution, and then studies some properties of these operators. At last, fourteen filters theorem in poset matroids is proved.2. Construtive operations of fuzzy independence set systems are studied. We take an example to say that some aspects are no relations between fuzzy independence set systems and its restriction,such as regularity and fundamental sequences. Vertical truncation, horizontal truncation and vertical restriction of fuzzy matroids are defined. we prove that the fundamental sequences of fuzzy matroids include the fundamental sequences of its restriction, and this is not true to fuzzy independence set systems. Fuzzy circuit transmission theorem is proved, on which we may study the connection of fuzzy matroids.3. A equivalent condition of closed fuzzy matroids is represented, it indicates the connection of HFM and fuzzy matroids. And then, induced space of fuzzy matroids is defined. We proved that a fuzzy matroid is closed if and only if its induced space is sequentially compact space.