## Dissertations - ALL

### Applications of Powerset Operators, Especially to Matroids

Spring 5-15-2022

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

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.

Open Access

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