Recently Hochstättler and Nesetril introduced the flow lattice of an oriented matroid as generalization of the lattice of all integer flows of a digraph or more general a regular matroid. This lattice is defined as the integer hull of the characteristic vectors of signed circuits. Here, we characterize the flow lattice of oriented matroids that are uniform or have rank 3 with a particular focus on the dimension of the lattice and construct a basis consisting of directed circuits. For general oriented matroids we introduce a 2-sum and decompose oriented matroids into 3-connected parts. We show how to determine the dimension of the lattice of 2-sums and conclude with some questions based on extensive experiments on small oriented matroids with connectivity at least 3.

Dedicated to Lex Schrijver on the occasion of his sixtieth birthday. There exist several theorems which state that when a matroid is representable over distinct fields � 1,..., � k, it is also representable over other fields. We prove a theorem, the Lift Theorem, that implies many of these results. First, parts of Whittle’s characterization of representations of ternary matroids follow from our theorem. Second, we prove the following theorem by Vertigan: if a matroid is representable over both GF(4) and GF(5), then it is representable over the real numbers by a matrix such that the absolute value of the determinant of every nonsingular square submatrix is a power of the golden ratio. Third, we give a characterization of the 3-connected matroids having at least two inequivalent representations over GF(5). We show that these are representable over the complex numbers. Additionally we provide an algebraic construction that, for any set of fields � 1,..., � k, gives the best possible result that can be proven using the Lift Theorem. 1

This thesis considers circular flow-type and circular chromatic-type parameters (φ and χ, respectively) for matroids. In particular we focus on orientable matroids and 6√1-matroids. These parameters are obtained via two approaches: algebraic and orientation-based. The general questions we discuss are: bounds for flow number; characterizations of Eulerian and bipartite matroids; and possible connections between the two possible extensions of φ: algebraic and orientation. In the case of orientable matroids, we obtain characterizations of bipartite rank-3 matroids and Eulerian uniform, rank-3 matroids; an asymptotic result regarding the flow number of uniform matroids; and an improvement on the known bound for flow number of matroids of arbitrary rank. This bound is further improved for the uniform case. For 6 √ 1-matroids, we examine an algebraic extension of the parameters χ and φ. We also

We show that there are exactly 2 k−1 Dk inequivalent orientations of the rank-k free spike (k ≥ 4) where Dk is the k th Dedekind number. Our proof of this result is constructive and employs the monotone boolean functions.

ABSTRACT. We extend the notion of representation of a matroid to algebraic structures that we call skew partial fields. Our definition of such representations extends Tutte’s definition, using chain groups. We show how such representations behave under duality and minors, we extend Tutte’s representability criterion to this new class, and we study the generator matrices of the chain groups. An example shows that the class of matroids representable over a skew partial field properly contains the class of matroids representable over a skew field. Next, we show that every multilinear representation of a matroid can be seen as a representation over a skew partial field. Finally we study a class of matroids called quaternionic unimodular. We prove a generalization of the Matrix Tree theorem for this class. 1.