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.
","abstract_html":"<p>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}\\perpv and called the \\textit{orthogonal complement} of \\(\\mathcal{S}\\). One of the fundamental theorems of vector space theory states that, (\\mathcal{S}\\perpv)\\perpv 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.</p><p>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. </p><p>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.</p><p>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.</p>","abstract_has_math":true,"creators":["Uricchio, Nathan"],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Graver, Jack E."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-05-15T07:00:00Z","date_published":"2022-05-15T07:00:00Z","updated_at":"2026-07-24T04:56:04Z","subjects":["Circuits","Clutter","Duality","Matroid","Operator","Powerset","Mathematics","Physical Sciences and Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://surface.syr.edu/etd/1369","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Graver, Jack E."]},{"key":"dc:creator","label":"Author","values":["Uricchio, Nathan"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Circuits","Clutter","Duality","Matroid","Operator","Powerset","Mathematics","Physical Sciences and Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://surface.syr.edu/etd/1369"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["Applications of Powerset Operators, Especially to Matroids"]}]}],"canonical_facts":{"dc:contributor":["Graver, Jack E."],"dc:creator":["Uricchio, Nathan"],"dc:description.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.
"],"dc:identifier":["https://surface.syr.edu/etd/1369"],"dc:subject":["Circuits","Clutter","Duality","Matroid","Operator","Powerset","Mathematics","Physical Sciences and Mathematics"],"dc:title":["Applications of Powerset Operators, Especially to Matroids"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T04:56:04Z"}