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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Fundamental theorem of geometry
without the 1-to-1 assumption


Authors: Alexander Chubarev and Iosif Pinelis
Journal: Proc. Amer. Math. Soc. 127 (1999), 2735-2744
MSC (1991): Primary 51A15; Secondary 51A05, 51A45, 51A25, 51D15, 51D30, 51E15, 51N10, 51N15, 14P99, 05B25
Published electronically: April 23, 1999
MathSciNet review: 1657778
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Abstract | References | Similar Articles | Additional Information

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.


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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

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05280-6
PII: S 0002-9939(99)05280-6
Keywords: Fundamental theorem of geometry, affine space, affine transformation, semi-affine transformation, collineation, isomorphism, parallelism, incidence relations, projective transformation
Received by editor(s): June 21, 1996
Published electronically: April 23, 1999
Communicated by: Christopher Croke
Article copyright: © Copyright 1999 American Mathematical Society