THESIS
2016

ix, 93 pages : illustrations ; 30 cm

**Abstract**
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....[

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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.

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