Abstract.
K. Menger and G. Birkhoff recognized 70 years ago that lattice theory provides a framework for the development of incidence geometry (affine and projective geometry). We show in this article that lattice theory also provides a framework for the development of metric geometry (including the euclidean and classical non-euclidean geometries which were first discovered by A. Cayley and F. Klein). To this end we introduce and study the concept of a Cayley–Klein lattice. A detailed investigation of the groups of automorphisms and an algebraic characterization of Cayley–Klein lattices are included.
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Received August 29, 2006; accepted in final form July 25, 2007.
The authors would like to thank an unknown referee for his helpful suggestions.
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Struve, H., Struve, R. Lattice theory and metric geometry. Algebra univers. 58, 461–477 (2008). https://doi.org/10.1007/s00012-008-2081-1
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DOI: https://doi.org/10.1007/s00012-008-2081-1