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Geometric Mappings on Geometric Lattices

Published online by Cambridge University Press:  20 November 2018

David Sachs
David Sachs*
Affiliation:
Wright State University, Dayton, Ohio
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It is a classical result of mathematics that there is an intimate connection between linear algebra and projective or affine geometry. Thus, many algebraic results can be given a geometric interpretation, and geometric theorems can quite often be proved more easily by algebraic methods. In this paper we apply topological ideas to geometric lattices, structures which provide the framework for the study of abstract linear independence, and obtain affine geometry from the mappings that preserve the closure operator that is associated with these lattices. These mappings are closely connected with semi-linear transformations on a vector space, and thus linear algebra and affine geometry are derived from the study of a certain closure operator and mappings which preserve it, even if the “space” is finite.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 1 , February 1971 , pp. 22 - 35
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Baer, R., Linear algebra and projective geometry (Academic Press, New York, 1952).Google Scholar
2. Birkhoff, G., Lattice theory, 3rd ed. (Amer. Math. Soc, Providence, R. I., 1967).Google Scholar
3. Crapo, H. and Rota, G.-C., Combinatorial geometries, Lecture notes, University of Waterloo and Massachusetts Institute of Technology, 1968.Google Scholar
4. Dubreil-Jacotin, M. L., Lesieur, L., and Croisot, R., Leçonssur la théorie des treillis des structuresalgébriquesordonnées et des treillisgéométriques(Gauthier-Villars, Paris, 1953).Google Scholar
5. Jónsson, B., Lattice-theoretic approach to projective geometry. The axiomatic method, with special reference to geometry and physics, Proc. Internat.Sympos.held at the Univ. of Calif., Berkeley, December 26, 1957-January 4, 1958, edited by Henkin, L., Suppes, P., and Tarski, A., pp. 188–203, Studies in logic and the foundations of mathematics (North-Holland, Amsterdam, 1959).Google Scholar
6. Wilcox, L. R., Extensions of semi-modular lattices. III, Bull. Amer. Math. Soc. 48 (1942), 35.Google Scholar