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Abstract. In this paper, derivable nets are described algebraically with the help of linear mappings. This gives a new approach to the derivation of derivable desarguesian a½ne planes that can be generalized to a½ne spaces of higher dimension. The resulting derived spaces are proper Sperner spaces. 2000 Mathematics Subject Classi®cation: 51A15, 51A25, 51A45. In [16] N. L. Johnson showed that every derivable net can be represented in a 3dimensional projective space, where the points of the net are exactly those lines of the projective space that are skew to a ®xed line. From this we obtain an algebraic description of derivable nets in terms of the endomorphism ring of a 2-dimensional left vector space. We use this in order to embed derivable nets in desarguesian a½ne planes and study under which conditions the planes then are derivable with respect to the net. There are two ways how to generalize our approach to the endomorphism rings of certain higher dimensional vector spaces. Hence we obtain two methods of ``derivation of n-dimensional a½ne spaces''. The derived spaces then are Sperner spaces and at the same time central translation structures; in case n b 2 they never are a½ne spaces. In
Forum Mathematicum – de Gruyter
Published: Jul 29, 2002
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