Date of Award

Spring 5-15-2022

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Advisor(s)

Graver, Jack E.

Keywords

Circuits, Clutter, Duality, Matroid, Operator, Powerset

Subject Categories

Mathematics | Physical Sciences and Mathematics

Abstract

Let \(\mathcal{V}\) denote a vector space over an arbitrary field with an inner product. For any collection \(\mathcal{S}\) of vectors from \(\mathcal{V}\) the collection of all vectors orthogonal to each vector in \(\mathcal{S}\) is a subspace, denoted as \(\mathcal{S}^{\perp_v}\) and called the \textit{orthogonal complement} of \(\mathcal{S}\). One of the fundamental theorems of vector space theory states that, \((\mathcal{S}^{\perp_v})^{\perp_v}\) is the subspace \textit{spanned} by \(\mathcal{S}\). Thus the ``spanning'' operator on the subsets of a vector space is the square of the ``orthogonal complement'' operator.

In matroid theory, the orthogonal complement of a matroid \(M\) is also well-defined and similarly results in another matroid. Although this new matroid is more commonly referred to as the `dual matroid', denoted as \(M^*\), and typically formed using a very different approach. There is an interesting relation between the circuits of a matroid \(M\) and the cocircuits of \(M\) (the circuits of its dual matroid \(M^*\)) which aligns much more closely to the orthogonal complement of a vector space.

We expand on this relation to define a powerset operator: \((\phantom{S})^*\). Given \(\mathcal{S} \subseteq \mathcal{P}(E)\), we denote \(\mathcal{S}^*\) to be the minimal sets of \(\{X \subseteq E \colon X \text{ is nonempty, } |X \cap A| \neq 1 \text{ for each } A \in \mathcal{S}\}\). We call this powerset operator the \textbf{circuit duality operator}. Unlike the vector space orthogonal complement operator, this circuit duality operator may not behave as nicely when applied to collections that do not correspond to a matroid.

This thesis is an investigation into the development of tools and additional operators to help understand the collections of sets that result in a matroid under one or more applications of the circuit duality operator.

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