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

By Alexander Chubarev, Iosif Pinelis and Communicated Christopher Croke


Abstract. It is proved that any mapping of an n-dimensional affine space over a division ring D onto itself which maps every line into a line is semiaffine, if n ∈{2,3,...} and D = Z2. 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 D other than R, Zp, orC – division rings. Terminology and notation Let D be a division ring and let L be a finite-dimensional linear space over D. An affine space [in L] overDis defined here simply as any set of the form A: = P +Λ, where P ∈Land Λ is a linear subspace of L. GivenLand A, the linear subspace Λ is uniquely determined; let us then put Tan A: = Λ and dim A: = dim Tan A. If A and Π are affine spaces [in L] overDand Π ⊆A, then Π is called an affine subspace of A. For brevity, let us refer to a k-dimensional affine space over D as a k-plane; then, 1-planes will be referred to simply as lines. In what follows, A is a

Year: 1999
OAI identifier: oai:CiteSeerX.psu:
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