Abstract
Let (P, \(\mathfrak{L}\)) and (P′, \(\mathfrak{L}\)′) be linear spaces satisfying the exchange axiom with dim P=dim P′ ∈ ℕ. Then a bijection ϕ:P→P′ which maps collinear points onto collinear points is an isomorphism. Also a surjection ψ:P→P′ which maps any three non-collinear points to non-collinear points is an isomorphism. This assertion is not true if dim P is not finite.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Benz, W.: Geometrische Transformationen unter besonderer Berücksichtigung der Lorentztransformationen. BI-Wissenschaftsverlag, Mannheim, Wien, Zürich, 1992.
Buekenhout, F.: Questions about linear spaces, Discrete Math. 129 (1994), 19–27.
Carter, D. and Vogt, A.: Collinearity-preserving functions between Desarguesian planes, Mem. Amer. Math. Soc. 235 (1980).
Ceccherini, P. V.: Collineazioni e semicollineazioni tra spazi affini o proiettivi, Rend. Mat. Appl. (7) 26 (1967), 309–348.
Huang, W.-L. and Kreuzer, A.: Basis preserving maps of linear spaces, Arch. Math. 64 (1995), 530–533.
Hughes, D. R. and Piper, F. C.: Projective Planes, Springer, New York, Heidelberg, Berlin, 1973.
Karzel, H., Sörensen, K. and Windelberg, D.: Einf:uhrung in die Geometrie, Göttingen, 1973.
Schröder, E. M.: Vorlesungen über Geometrie, Band 2. Affina und projektive Geometrie, BI-Wissenschaftsverlag, Mannheim, Wien, Zürich, 1991.
Sörensen, K.: Der Fundamentalsatz für Projektionen, Mitt. Math. Ges. Hamburg XI (1985), 303–309.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kreuzer, A. On the definition of isomorphisms of linear spaces. Geom Dedicata 61, 279–283 (1996). https://doi.org/10.1007/BF00150028
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00150028