Planar and affine spaces |
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Authors: | P?nar Anapa,?brahim Gü nalt?l? |
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Affiliation: | a Education Faculty, Primary Education Department, Eskisehir Osmangazi University, 26480, Eskisehir, Turkey b Mathematics Department, Eskisehir Osmangazi University, 26480, Eskisehir, Turkey c Department of Pure Mathematics and Computer Algebra, Ghent University, Krijgslaan 281, S22, 9000 Gent, Belgium |
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Abstract: | In this note, we characterize finite three-dimensional affine spaces as the only linear spaces endowed with set Ω of proper subspaces having the properties (1) every line contains a constant number of points, say n, with n>2; (2) every triple of noncollinear points is contained in a unique member of Ω; (3) disjoint or coincide is an equivalence relation in Ω with the additional property that every equivalence class covers all points. We also take a look at the case n=2 (in which case we have a complete graph endowed with a set Ω of proper complete subgraphs) and classify these objects: besides the affine 3-space of order 2, two small additional examples turn up. Furthermore, we generalize our result in the case of dimension greater than three to obtain a characterization of all finite affine spaces of dimension at least 3 with lines of size at least 3. |
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Keywords: | Planar spaces Affine spaces Linear spaces |
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