(Stellenbosch : Stellenbosch University, 2022-12) Rabie, Jacques; Howell, Karin-Therese; Stellenbosch University. Faculty of Science. Dept. of Applied Mathematics.

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ENGLISH 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.