## Contributions to the theory of near vector spaces

##### Abstract

The main purpose of this thesis is to give an exposition of and expand the theory of near vector spaces, as first introduced by Andr´e [1]. The notion of a vector space is well known. For this reason the material in this thesis is presented in such a way that the parallels between near vector spaces and vector spaces are apparent. In Chapter 1 several elementary definitions and properties are given. In addition, some important examples that will be referred to throughout this paper are cited. In Chapter 2 the theory of near vector spaces is presented. We start off with some preliminary results in 2.1 and build up to the definition of a regular near vector space in 2.5. In addition, we show how a near vector space can be decomposed into maximal regular subspaces. We conclude this chapter by showing when a near vector space will in fact be a vector space. We follow the format of De Bruyn’s thesis; however, both De Bruyn and Andr´e make use of left nearfields to define the near vector spaces. In light of the material we want to present in Chapter 4, it is more standard to use the notation as in the papers by van der Walt, [12], [13]. Thus we develop the material using right nearfields with scalar multiplication on the right of vectors. The third chapter contains some examples of near vector spaces and serves as an illustration of much of the work of Chapter 2. Examples 1, 2 and 3 were used in De Bruyn’s thesis. However, on closer inspection, it was revealed that in Example 2, the element (a, 0, 0, d) is omitted as an element of Q(V ). This error is corrected. And in keeping with our use of right nearfields, the necessary changes are made to Example 1 and 3. In particular, the definition of ◦ in Example 3 is adapted and the necessary adjustments are made. We conclude this chapter by developing a theory that allows us to characterise all finite dimensional near vector spaces over Zp, for p a prime. In Chapter 4 we turn our attention to the work done by van der Walt in [12] and [13]. In Section 4.1 we consider the effects that ‘perturbations’ in the action of a (right) nearfield F has on the well known structures, the ring of linear transformations of V and the nearring of homogeneous functions of V into itself. This first section sets the scene for the more generalised situation described in 4.2 and leads to the introduction of the nearring of matrices determined by n multiplicatively isomorphic nearfields and a matrix of isomorphisms. We conclude this chapter by summarising some properties of this nearring in 4.3 and 4.4. Note that throughout this paper, ⊂ will be used to convey a proper subset, whereas ⊆ will convey the possibility of equality.

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