Abstract
The structure of lines in a projective or a polar Grassmann space together with binary coplanarity is a sufficient system of primitive notions for these geometries.
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Batten L.M.: Locally generalized projective lattices satisfying a Bundle Theorem, Geom. Dedicata 22, 363–369 (1987)
Benz, W.: Geometrische Transformationen, B. I. Wiessenschaftsverlag, Mannheim, Leipzig, Wien, Zürich (1992)
Benz W., Schröder E.M.: Bestimmung der orthogonalitätstreuen Permutationen euklidischer Räume. Geom. Dedicata 21(3), 263–276 (1986)
Brauner H.: Eine Kennzeichnung der Ähnlichkeiten affiner R äume mit definiter Orthogonalitätsstruktur. Geom. Dedicata 29(1), 45–51 (1989)
Brauner H.: Über die von Kollineationen projektiver R äume induzierten Geradenabbildungen. Sitz. Ber. Österr. Akad. Wiss., Math.-Natur. Kl. Sitzungsber. II. 197(4–7), 327–332 (1988)
Chow W.-L.: On the geometry of algebraic homogeneous spaces. Ann. Math. 50, 32–67 (1949)
Cohen M.A.: Point-line spaces related to buildings. In: Buekenhout, F. (ed.) Handbook of Incidence Geometry, pp. 647–737. North-Holland, Amsterdam (1995)
Havlicek H.: A characteristic property of elliptic Plücker transformations. J. Geom 58(1–2), 106–116 (1997)
Havlicek, H.: Isomorphisms of affine Plücker spaces. In: Artémiadis N.K., Stephanidis N.K. (eds.) Proceedings of the 4th International Congress of Geometry (Thessaloniki, 1996), pp. 171–178. Giachoudis-Giapoulis, Thessaloniki (1997)
Havlicek H.: On isomorphisms of Grassmann spaces. Mitt. Math. Ges. Hambg. 14, 117–120 (1995)
Havlicek H.: On Plücker transformations of generalized elliptic spaces. Rend. Mat. Appl. 7(15), 39–56 (1994)
Havlicek H.: Symplectic Plücker transformations. Math. Pannon. 6, 145–153 (1995)
Havlicek H., Pambuccian V.: On the axiomatics of projective and affine geometry in terms of line intersection. Result Math. 45, 35–44 (2004)
Hedrick E.R., Ingold L.: A set of axioms for line geometry. Trans. Am. Math. Soc. 15, 205–214 (1914)
Kahn J.: Locally projective-planar lattices which satisfy Bundle Theorem. Math Z. 175, 219–247 (1980)
Koźniewski E.: An axiom system for the line geometry. Demonstr. Math. 21, 1071–1087 (1988)
Kramer R.: The undefinability of intersection from perpendicularity in the three-dimensional Euclidean geometry of lines. Geom. Dedicata 46(2), 207–210 (1993)
Kreuzer A.: Locally projective spaces which satisfy the Bundle Theorem. J. Geom. 56, 87–98 (1996)
List K.: Harmonic mappings and hyperbolic Plücker transformations. J. Geom. 70(1–2), 108–116 (2001)
List K.: On orthogonality-preserving Plücker transformations of hyperbolic spaces. Abh. Math. Sem. Univ. Hambg. 70, 63–75 (2000)
Pambuccian V.: An axiomatics for hyperbolic projective-metric planes in terms of lines and orthogonality. Bull. Pol. Acad. Sci. Math. 52(3), 297–302 (2004)
Pambuccian V.: Aufbau der hyperbolischen Geometrie aus dem Geradenorthogonalitätsbegriff. Acta Math. Hung. 101(1–2), 51–61 (2003)
Pambuccian, V.: Elementary axiomatizations of projective space and its associated Grassmann space. Note Mat. 24(1), 129–141 (2004/2005)
Pambuccian V., Pra zmowski K., Sakowicz K.: Defining co-punctuality in terms of line-orthogonality in plane hyperbolic geometry. Acta Math. Hung. 109(4), 289–293 (2005)
Pankov M.: Grassmannians of classical buildings Algebra and Discrete Mathematics Vol. 2. World Scientific, New Jersey (2010)
Pankov M., Pra zmowski K., Zynel M.: Geometry of polar Grassmann spaces. Demonstr. Math. 39(3), 625–637 (2006)
Pieri, M.: Sui principi che regono la geometria delle rette. Atti Accad. Torino 36, 335–350 (1901). (Also in: Opere scelte, Opere sui fondamenti della matematica, pp. 273–288, UMI, Bologna 1980)
Pra zmowski K., Zynel M.: Orthogonality of subspaces in metric-projective geometry. Adv. Geom. 11(1), 103–116 (2011)
Pra zmowski K., Zynel M.: Possible primitive notions for geometry of spine spaces. J. Appl. Logic 8(3), 262–276 (2010)
Stoß H.-J.: Treffgeraden und Nullinvarianz. Beiträge zur Liniengeometrie. Philosophisch-Anthroposophischer Verlag am Goetheanum, Dornach (1995)
Schwabhäuser W., Szczerba L.W.: Relations on lines as primitive notions in Euclidean geometry. Fundam. Math. 82, 347–355 (1975)
Tallini G.: Partial line spaces and algebraic varieties. Symp. Math. 28, 203–217 (1986)
Trott S.: An axiomatic line geometry. Can. J. Math. 20, 939–952 (1968)
Winternitz A.: Zur Begründung der projektiven Geometrie: Einführung idealer Elemente unabhänging von der Anordnung. Ann. Math. (Second Series) 41, 365–390 (1940)
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Petelczyc, K., Żynel, M. Coplanarity of lines in projective and polar Grassmann spaces. Aequat. Math. 90, 607–623 (2016). https://doi.org/10.1007/s00010-015-0387-6
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DOI: https://doi.org/10.1007/s00010-015-0387-6