Fundamental theorem of geometry without the 1-to-1 assumption
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- by Alexander Chubarev and Iosif Pinelis PDF
- Proc. Amer. Math. Soc. 127 (1999), 2735-2744 Request permission
Abstract:
It is proved that any mapping of an $n$-dimensional affine space over a division ring $\mathbb {D}$ onto itself which maps every line into a line is semi-affine, if $n\in \{2,3,\dots \}$ and $\mathbb {D}\ne \mathbb {Z}_{2}$. This result seems to be new even for the real affine spaces. Some further generalizations are also given. The paper is self-contained, modulo some basic terms and elementary facts concerning linear spaces and also – if the reader is interested in $\mathbb {D}$ other than $\mathbb {R}$, $\mathbb {Z}_{p}$, or $\mathbb {C}$ – division rings.References
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Additional Information
- Alexander Chubarev
- Affiliation: Cimatron Ltd., Gush Etzion 11, Givat Shmuel, 54030, Israel
- Email: sasha@cimatron.co.il
- Iosif Pinelis
- Affiliation: Department of Mathematical Sciences, Michigan Technological University, Hough- ton, Michigan 49931
- Email: ipinelis@math.mtu.edu
- Received by editor(s): June 21, 1996
- Published electronically: April 23, 1999
- Communicated by: Christopher Croke
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 2735-2744
- MSC (1991): Primary 51A15; Secondary 51A05, 51A45, 51A25, 51D15, 51D30, 51E15, 51N10, 51N15, 14P99, 05B25
- DOI: https://doi.org/10.1090/S0002-9939-99-05280-6
- MathSciNet review: 1657778