Contributions to the theory of near-vector spaces, their geometry, and hyperstructures

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dc.contributor.advisorHowell, Karin-Thereseen_ZA
dc.contributor.authorRabie, Jacquesen_ZA
dc.contributor.otherStellenbosch University. Faculty of Science. Dept. of Applied Mathematics.en_ZA
dc.date.accessioned2022-10-25T19:51:56Z
dc.date.accessioned2023-01-16T12:41:12Z
dc.date.available2022-10-25T19:51:56Z
dc.date.available2023-01-16T12:41:12Z
dc.date.issued2022-12
dc.descriptionThesis (PhD)--Stellenbosch University, 2022.en_ZA
dc.description.abstractENGLISH ABSTRACT: This thesis expands on the theory and application of near-vector spaces — in particular, the underlying geometry of near-vector spaces is studied, and the theory of near-vector spaces is applied to hyperstructures. More specifically, a near-linear space is defined and some properties of these spaces are proved. It is shown that by adding some axioms, the nearaffine space, as defined by André, i s obtained. A correspondence is shown between subspaces of nearaffine spaces generated by near-vector spaces, and the cosets of subspaces of the corresponding near-vector space. As a highlight, some of the geometric results are used to prove an open problem in near-vector space theory, namely that a non-empty subset of a near-vector space that is closed under addition and scalar multiplication is a subspace of the near-vector space. The geometric work of this thesis is concluded with a first look into the projections of nearaffine s paces, a branch of the geometry that contains interesting avenues for future research. Next the theory of hyper near-vector spaces is developed. Hyper near-vector spaces are defined having similar properties to André’s near-vector space. Important concepts, including independence, the notion of a basis, regularity, and subhyperspaces are defined, and an analogue of the Decomposition Theorem, an important theorem in the study of near-vector spaces, is proved for these spaces.en_ZA
dc.description.abstractAFRIKAANS OPSOMMING: Hierdie tesis bou op die teorie en toepassing van naby-vektorruimtes — besonderlik word die onderliggende meetkunde van naby-vektorruimtes bestudeer en die teorie van naby-vektorruimtes word toegepas op hiperstrukture. Spesifiek work ’n naby-lineêre ruimte gedefinieer en sommige eienskappe van hier- die ruimtes word bewys. Dit word bewys dat, deur sekere aksiomas by te las, die naby-affiene ruimte, soos gedefinieer deur André, verkry w ord. ’n Verwantskap tus- sen die deelruimtes van naby-affiene ruimtes gegenereer deur naby-vektorruimtes en die resklasse van die deelruimtes van die verwante naby-vektorruimte word be- wys. As ’n hoogtepunt word van die meetkundige resultate gebruik om ’n oop probleem op te los in naby-vektorruimteteorie, naamlik dat ’n nie-leë deelversa- meling van ’n naby-vektorruimte wat geslote is onder optelling en skalaarverme- nigvuldiging ’n deelruimte is van die naby-vektorruimte. Die meetkundige werk in dié tesis sluit af met ’n eerste bestudering van projeksies van naby-affiene ruimtes, ’n tak in die meetkunde wat interessante toekomstige navorsingsrigtings bevat. Volgende word die teorie agter hiper naby-vektorruimtes ontwikkel. Hiper naby- vektorruimtes word gedefinieer s oortgelyk a an A ndré s e n aby-vektorruimte. Be- langrike konsepte, insluitent onafhanklikheid, die begrip van ’n basis, regulêriteit en hiper-deelruimtes word gedefinieer e n ’ n analoog van die Ontbindingstelling, belangrik in die teorie van naby-vektorruimtes, word bewys vir hierdie ruimtes.af_ZA
dc.description.versionDoctoralen_ZA
dc.format.extentvi, 84 pagesen_ZA
dc.identifier.urihttp://hdl.handle.net/10019.1/125917
dc.language.isoen_ZAen_ZA
dc.publisherStellenbosch : Stellenbosch University
dc.rights.holderStellenbosch Universityen_ZA
dc.subjectNear-vector spacesen_ZA
dc.subjectNearaffine spacesen_ZA
dc.subjectIncidence geometry -- Mathematical modelsen_ZA
dc.subjectHypergroupsen_ZA
dc.subjectHyper near-vector spacesen_ZA
dc.subjectGeometry, Differentialen_ZA
dc.subjectDecomposition theorem -- Mathematical modelsen_ZA
dc.subjectUCTDen_ZA
dc.titleContributions to the theory of near-vector spaces, their geometry, and hyperstructuresen_ZA
dc.typeThesisen_ZA
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