{"id":{"repo_id":"the-open-u","oai_identifier":"oai:oro.open.ac.uk:56903"},"canonical_url":"https://search.dev.ndltd.org/etd/the-open-u/oai:oro.open.ac.uk:56903","repository":{"repo_id":"the-open-u","name":"The Open University","base_url":"https://oro.open.ac.uk/cgi/oai2"},"display":{"title":"Theory and applications of freedom in matroids","abstract":"To each cell e in a matroid M we can associate a non-negative integer ǁ e ǁ called the freedom of e. Geometrically the value ǁ e ǁ indicates how freely placed the cell is in the matroid. We see that ǁ e ǁ is equal to the degree of the modular cut generated by all the fully-dependent flats of M containing e. The relationship between freedom and basic matroid constructions, particularly one-point lifts and duality, is examined, and the applied to erections. We see that the number of times a matroid M can be erected is related to the degree of the modular cut generated by all the fully-dependent flats of M*. If ζ(M) is the set of integer polymatroids with underlying matroid structure M, then we show that for any cell e of M ǁ e ǁ= \\frac{max\\ f \\ (e)}{f\\in\\zeta} We look at freedom in binary matroids and show that for a connected binary matroid M, ǁ e ǁ is the number of connected components of M/e. Finally the matroid join is examined and we are able to solve a conjecture of Lovasz and Recski that a connected binary matroid M is reducible if and only if there is a cell e of M with M/e disconnected.","abstract_html":"To each cell e in a matroid M we can associate a non-negative integer ǁ e ǁ called the freedom of e. Geometrically the value ǁ e ǁ indicates how freely placed the cell is in the matroid. We see that ǁ e ǁ is equal to the degree of the modular cut generated by all the fully-dependent flats of M containing e. The relationship between freedom and basic matroid constructions, particularly one-point lifts and duality, is examined, and the applied to erections. We see that the number of times a matroid M can be erected is related to the degree of the modular cut generated by all the fully-dependent flats of M*. If ζ(M) is the set of integer polymatroids with underlying matroid structure M, then we show that for any cell e of M ǁ e ǁ= \\frac{max\\ f \\ (e)}{f\\in\\zeta} We look at freedom in binary matroids and show that for a connected binary matroid M, ǁ e ǁ is the number of connected components of M/e. Finally the matroid join is examined and we are able to solve a conjecture of Lovasz and Recski that a connected binary matroid M is reducible if and only if there is a cell e of M with M/e disconnected.","abstract_has_math":false,"creators":["Duke, R."],"institution":"The Open University","degree_name":"phd","degree_level":"doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1981,"date_issued":"1981","date_published":"1981","updated_at":"2026-07-24T05:02:56Z","subjects":[],"languages":["en"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":null,"outbound_label":null,"outbound_source":null},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Duke, R."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["1981"]},{"key":"dc:date.issued","label":"Date","values":["1981"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["The Open University"]},{"key":"dc:relation.isreferencedby","label":"Dc Relation Isreferencedby","values":["https://oro.open.ac.uk/56903/"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["phd"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://oro.open.ac.uk/56903/1/354270.pdf"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["To each cell e in a matroid M we can associate a non-negative integer ǁ e ǁ called the freedom of e. Geometrically the value ǁ e ǁ indicates how freely placed the cell is in the matroid. We see that ǁ e ǁ is equal to the degree of the modular cut generated by all the fully-dependent flats of M containing e. The relationship between freedom and basic matroid constructions, particularly one-point lifts and duality, is examined, and the applied to erections. We see that the number of times a matroid M can be erected is related to the degree of the modular cut generated by all the fully-dependent flats of M*. If ζ(M) is the set of integer polymatroids with underlying matroid structure M, then we show that for any cell e of M ǁ e ǁ= \\frac{max\\ f \\ (e)}{f\\in\\zeta} We look at freedom in binary matroids and show that for a connected binary matroid M, ǁ e ǁ is the number of connected components of M/e. Finally the matroid join is examined and we are able to solve a conjecture of Lovasz and Recski that a connected binary matroid M is reducible if and only if there is a cell e of M with M/e disconnected."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Theory and applications of freedom in matroids"]}]}],"canonical_facts":{"dc:creator":["Duke, R."],"dc:date":["1981"],"dc:date.issued":["1981"],"dc:description.abstract":["To each cell e in a matroid M we can associate a non-negative integer ǁ e ǁ called the freedom of e. Geometrically the value ǁ e ǁ indicates how freely placed the cell is in the matroid. We see that ǁ e ǁ is equal to the degree of the modular cut generated by all the fully-dependent flats of M containing e. The relationship between freedom and basic matroid constructions, particularly one-point lifts and duality, is examined, and the applied to erections. We see that the number of times a matroid M can be erected is related to the degree of the modular cut generated by all the fully-dependent flats of M*. If ζ(M) is the set of integer polymatroids with underlying matroid structure M, then we show that for any cell e of M ǁ e ǁ= \\frac{max\\ f \\ (e)}{f\\in\\zeta} We look at freedom in binary matroids and show that for a connected binary matroid M, ǁ e ǁ is the number of connected components of M/e. Finally the matroid join is examined and we are able to solve a conjecture of Lovasz and Recski that a connected binary matroid M is reducible if and only if there is a cell e of M with M/e disconnected."],"dc:format":["application/pdf"],"dc:identifier.uri":["https://oro.open.ac.uk/56903/1/354270.pdf"],"dc:language":["en"],"dc:publisher.institution":["The Open University"],"dc:relation.isreferencedby":["https://oro.open.ac.uk/56903/"],"dc:title":["Theory and applications of freedom in matroids"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["doctoral"],"dc:type.qualificationname":["phd"]},"updated_at":"2026-07-24T05:02:56Z"}