The subspace structure of Beidleman near-vector spaces is investigated. We characterise finite dimensional Beidleman near-vector spaces and we classify the R-subgroups of finite dimensional Beidleman near-vector spaces. We provide an algorithm to compute the smallest R-subgroup containing a given set of vectors. Finally, we classify the subspaces of finite dimensional Beidleman near-vector spaces.
For any nearfield (R, +, •), denote by D(R) the set of all distributive elements of R. Let R be a finite Dickson nearfield that arises from Dickson pair (q, n). For a given pair (α, β) ∈ R 2 we study the generalized distributive set•" is the multiplication of the Dickson nearfield. We find that D(α, β) is not in general a subfield of the finite field F q n . In contrast to the situation for D(R), we also find that D(α, β) is not in general a subnearfield of R. We obtain sufficient conditions on α, β for D(α, β) to be a subfield of F q n and derive an algorithm that tests if D(α, β) is a subfield of F q n or not. We also study the notions of R-dimension, R-basis, seed sets and seed number of R-subgroups of the Beidleman near-vector spaces R m where m is a positive integer. Finally we determine the maximal R-dimension of gen(v 1 , v 2 ) for v 1 , v 2 ∈ R m , where gen(v 1 , v 2 ) is the smallest R-subgroup containing the vectors v 1 and v 2 .
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