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Extended dual affine planes

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Abstract

We study a class of diagram geometries, achieve a characterization of extended dual affine planes, and embed extended dual affine planes in extended projective planes. The geometries studied are rank 3 diagram geometries such that the residue of a point is a dual net, and the residue of a plane is linear; the dual of such a geometry has partitions on lines and planes which are reminiscent of parallelism of lines and planes of an affine 3-space. Examples of these geometries (some in dual form) include extended dual affine planes, Laguerre planes, 3-nets, and orthogonal arrays of strength 3. Theorem: Any such finite geometry satisfying Buekenhout's intersection property, and such that any two points are coplanar, is an extended dual affine plane (and has order 2, 4, or 10). Theorem: This geometry may be embedded in an extended projective plane of the same order.

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Authors and Affiliations

  1. Department of Computer and Information Science, Ohio State University, 43210, Columbus, OH, U.S.A.

    Alan P. Sprague

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  1. Alan P. Sprague
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This research was partially supported by NSF Grant MCS-8102361.

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Sprague, A.P. Extended dual affine planes. Geom Dedicata 16, 107–121 (1984). https://doi.org/10.1007/BF00147426

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  • DOI: https://doi.org/10.1007/BF00147426

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