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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Projective geometry on partially ordered sets


Authors: Ulrich Faigle and Christian Herrmann
Journal: Trans. Amer. Math. Soc. 266 (1981), 319-332
MSC: Primary 51A05; Secondary 06C05
DOI: https://doi.org/10.1090/S0002-9947-1981-0613799-9
MathSciNet review: 613799
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Abstract: A set of axioms is presented for a projective geometry as an incidence structure on partially ordered sets of "points" and "lines". The axioms reduce to the axioms of classical projective geometry in the case where the points and lines are unordered. It is shown that the lattice of linear subsets of a projective geometry is modular and that every modular lattice of finite length is isomorphic to the lattice of linear subsets of some finite-dimensional projective geometry. Primary geometries are introduced as the incidence-geometric counterpart of primary lattices. Thus the theory of finite-dimensional projective geometries includes the theory of finite-dimensional projective Hjelmslev-spaces of level $ n$ as a special case. Finally, projective geometries are characterized by incidence properties of points and hyperplanes.


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DOI: https://doi.org/10.1090/S0002-9947-1981-0613799-9
Keywords: Point, line, incidence structure, projective geometry, modular lattice
Article copyright: © Copyright 1981 American Mathematical Society

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