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 Bruynâs 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 Bruynâs
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 âperturbationsâ 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, â will be used to convey a proper subset, whereas â
will convey the possibility of equality.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:ufs/oai:etd.uovs.ac.za:etd-08252008-081056 |
| Date | 25 August 2008 |
| Creators | Howell, Karin Therese |
| Contributors | Prof JH Meyer |
| Publisher | University of the Free State |
| Source Sets | South African National ETD Portal |
| Language | en-uk |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | http://etd.uovs.ac.za//theses/available/etd-08252008-081056/restricted/ |
| Rights | unrestricted, I hereby certify that, if appropriate, I have obtained and attached hereto a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dissertation, or project report, allowing distribution as specified below. I certify that the version I submitted is the same as that approved by my advisory committee. I hereby grant to University Free State or its agents the non-exclusive license to archive and make accessible, under the conditions specified below, my thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report. |
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