The Categorical Aspects Of Matroid Theory

Integrating with different subjects is a typical property in the research of matroids, which makes matroid theory, a young mathematics branch, rich and strong. In this paper, we will focus on the categorical aspects of matroid theory, that is, we will study the fundemental properties of the category of matroids, and interpret some important results in matroid theory in terms of category. There are two kinds of relations in matroid theory which takes our attentions. One is the relation between simple finite matroids and finite geometric lattices, which is involved in the research of classical matroid theory (see [12]). It had been proved that there is a one to one correspondence between these two classes of objects. But what is their relation in categorical sense ? Whether the category of simple finite matroids and the category of finite geometric lattices is isomorphic or equivalent if the corresponding categories are established ? It is a new question worthy to be researched. The other is the relation between poset matroids and combinatorial schemes, which is involved in the research of poset matroids. Poset matroid is defined by replacing the ground set of a matroid with a finite poset and also combinatorial schemes are defined in a finite distributive lattices in [37,38], where the author called it the lattical counterparts of poset matroids. Can we find a one to one correspondence between these two ? Can we find appropriate maps to make the related categories well related ? To completely consider these questions can undoubtly provide some new thoughts and subjects. In this paper we aims to study the fundamental properties of the above mentioned categories and then explain the their relations accurately in the language of category theory. Fortunately, the strong maps on geometric lattices (called geometric homomorphism here) had been given by Higgs in [13] and Crapo in [26], and the strong maps on matroids had also been defined by Welsh in [12] aad Kung in [32]. Thus we guess they may be the ideal morphisms for the related categories. Although the category of matroids had been mentioned in some reference, it was just mentioned. The related references on this, especially on the maps of poset matroids or combinatorial schemes can hardly be found. Thus it will be a little difficult for us. What we have done may be meaningful in matroid theory, but it is just a start in category degree since there are still many important questionunsolved. Now we give the main content of this paper as follows:1. The category FMRS of finite matroids with rank strong maps and the category FMS of finite matroids with strong maps are studied systematicaly, and specially, the category SFMS of simple finite matroids and strong maps is proved to be a reflective full subcategory of FMS. Firstly, we point out some errors in [12] on the definition or properties of strong maps and weak maps, to correct them, we give several other definitions on matroids such as rank strong map (RS-map), map which reflects flats (RF-map), map which reflects independent sets (RI-map) and rank weak maps (RW-map). We point out that strong maps are essentially the rank strong map from the extended domain-matroid to the extended codomain-matroid. To make a strong map is weak, we redefine weak map. Secondly, the corresponding categories are naturally established and the fundamental properties of FMRS (the category of all finite matroids and rank strong maps) and the category FMS (the category of all finite matroids and strong maps) are studied systematicaly. We obtain that the restricted matroids are just subobjects and the contracted matroids are not necessarily subobject but quotient objects in FMS. The direct sum is just the coproducts but not the products. Especially, it is obtained that the category FMRS is not algebraic. Also some properties of FMW (the category of all finite matroids and weak maps) and FMRW (the category of all finite matroids and rank weak maps) are discussed. Finally, the simplifization of matroids is studied completely and SFMS (the category of all simple and finite matroids and strong maps) is proved to be a reflective full subcategory of FMS.2. Some fundamental properties of the categoty of geometric lattices and geometric lattice homomorphisms are studied, and the geometric representations of the previous categories of matroids are given, that is, they are shown to be equivalent with some subcategories of GL. Also the relations between the abstract categories of matroids and the corresponding concrete categories of matroids are cleared. Firstly, some characterizations of geometric lattice homomorphism are given and the equivalent definitons of sub-geometric lattices and quotient geometric lattices in [13] are given, which are just to be the subobjects and quotient objects in GL (the category of all geometric lattices and geometric lattice homomorphisms). Especially, the sub geometric lattices are the extremal subobjetcs and the quotient geometric lattices are the extremal quotient objects. Unlike in the general category of lattices, the finite caretesian products of geometric lattices are not GL-product but GL-...