In this thesis, we introduce the concepts of Z-closure operators and Z-flat lattices of arithmetic matroids, and show that for a representable arithmetic matroid, the characteristic polynomial of the associated toric arrangement is equal to the characteristic polynomial of the associated Z-flat at lattice. Also, we study the connectivity in arithmetic matroids. The main results are the characterizations of disconnected arithmetic matroids and the uniqueness of decompositions of arithmetic matroids.

The thesis consists of five chapters, which are organized as follows. In Chapter 1, we give a brief introduction to matroids and Tutte polynomials. Chapter 2 includes the theory of arithmetic matroids, arithmetic Tutte polynomials, and several other basic notions and constructions....[ Read more ]

In this thesis, we introduce the concepts of Z-closure operators and Z-flat lattices of arithmetic matroids, and show that for a representable arithmetic matroid, the characteristic polynomial of the associated toric arrangement is equal to the characteristic polynomial of the associated Z-flat at lattice. Also, we study the connectivity in arithmetic matroids. The main results are the characterizations of disconnected arithmetic matroids and the uniqueness of decompositions of arithmetic matroids.

The thesis consists of five chapters, which are organized as follows. In Chapter 1, we give a brief introduction to matroids and Tutte polynomials. Chapter 2 includes the theory of arithmetic matroids, arithmetic Tutte polynomials, and several other basic notions and constructions. In Chapter 3, we study the representable arithmetic matroids in detail and discuss the Tutte quasi-polynomials, which interpolate between the ordinary and the arithmetic Tutte polynomials. We show that evaluating the Tutte quasi-polynomial at some special points gives the number of elements of the complement of a certain group arrangement, and get the comparison formulae for the coefficients of the associated Tutte quasi-polynomials of some special representable arithmetic matroids. In Chapter 4, we introduce the notions of Z-closure operator and Z-flat lattice. One of the results is that the characteristic polynomial of an arithmetic matroid is equal to the characteristic polynomial of the associated Z-flat lattice, and if this arithmetic matroid is representable, then both of them are equal to the characteristic polynomial of the associated toric arrangement. Chapter 5 is devoted to the connectivity in arithmetic matroids. We give some characterizations of disconnected arithmetic matroids and completely disconnected arithmetic matroids. Also, we show that every arithmetic matroid can be decomposed into a direct sum of connected arithmetic matroids, which is unique up to weak isomorphism.