TAILIEUCHUNG - On spanning sets and generators of near-vector spaces

In this paper we study the quasi-kernel of certain constructions of near-vector spaces and the span of a vector. We characterize those vectors whose span is one-dimensional and those that generate the whole space. | Turk J Math (2018) 42: 3232 – 3241 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article On spanning sets and generators of near-vector spaces Karin-Therese HOWELL1∗,, Sogo Pierre SANON2 , Department of Mathematical Sciences, Faculty of Science, Stellenbosch University, Stellenbosch, South Africa 2 Department of Mathematical Sciences, Faculty of Science, Stellenbosch University, Stellenbosch, South Africa 1 Received: • Accepted/Published Online: • Final Version: Abstract: In this paper we study the quasi-kernel of certain constructions of near-vector spaces and the span of a vector. We characterize those vectors whose span is one-dimensional and those that generate the whole space. Key words: Field, nearfield, vector space, near-vector space 1. Introduction The near-vector spaces we study in this paper were first introduced by André in 1974 [1]. His near-vector spaces have less linearity than normal vector spaces. They have been studied in several papers, including [2–6]. More recently, since André did a lot of work in geometry, their geometric structure has come under investigation. In order to construct some incidence structures a good understanding of the span of a vector is necessary. It very quickly became clear that near-vector spaces exhibit some strange behavior, where the span of a vector need not be one-dimensional and it is possible for a single vector to generate the entire space. In this paper we begin by giving the preliminary material of near-vector spaces. In Section 3 we take a closer look at the class of near-vector spaces of the form (F n , F ) , where F is a nearfield and n is a natural number, constructed using van der Walt’s important construction theorem in [9] for finite dimensional nearvector spaces. We give conditions for when the quasi-kernel will be the whole space. In the last section we prove that when for a near-vector space .

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