Mathematics Wiskunde

Friday 14 October 2022

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

Speaker: Jacques Rabie
Time: 12:00
Venue: Mathematics Division, Room 3001 (streamed)

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.

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 spaces.

Next the theory of hyper near-vector spaces is developed. Hyper near-vector space are defined having similar properties to Andr ́e’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.

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