Mathematics Wiskunde

Nearfields and nearrings were first studied by Dickson in 1905 and were first applied in geometry. Despite appearing to be very similar to fields, the weakening of the one distributive law makes the study interesting. Several researchers, including Beidleman, Andre, Karzel and Wähling defined near-vector spaces in different ways. Our thesis is a first step towards a detailed study of Beidleman near-vector spaces, as first introduced by Beidleman in his thesis in 1966. As the main results of our thesis, we characterise the finite-dimensional Beidleman near-vector spaces and present an algorithm called EGE (Expanded Gaussian Elimination) which determines the smallest $$R$$-subgroup containing a given finite set of vectors in $$R^n)) where \(R$$ is a proper nearfield and n is a positive integer. We will also classify all the subspaces of $$R^n$$ by means of an algorithm called the Adjustment of the EGE algorithm. Finally, we will discuss the generalized distributive set of a finite nearfield.