Partial Representations for Ternary Matroids

In combinatorics, a matroid is a discrete object that generalizes various notions of dependence that arise throughout mathematics. All of the information about some matroids can be encoded (or represented) by a matrix whose entries come from a particular field, while other matroids cannot be represented in this way. However, for any matroid, there exists a matrix, called a partial representation of the matroid, that encodes some of the information about the matroid. In fact, a given matroid usually has many different partial representations, each providing different pieces of information about the matroid. In this thesis, we investigate when a given partial representation actually encodes all of the information about some matroid. In particular, we restrict our attention to the class of ternary matroids, which are those that can be fully represented by a matrix over the Galois field GF(3). We explore some of the conditions under which such a matroid is also uniquely determined by one of its partial representations