In a previous paper the author recapitulated betweenness geometry, developed in 1904–64 by O. Veblen, J. Sarv, J. Hashimoto, and the author. The relationship of this geometry with join geometry (by W. Prenowitz) was investigated. Now this relationship will be extended to convex and linear geometry. The achievements of the well-developed projective plane geometry are used to enrich betweenness plane geometry with coordinates, ternary operation, algebraic extension, Lenz–Barlotti classification, translation, and Moufang type. The final statement is that every Moufang-type betweenness plane is Desarguesian.
1. Gauss, C. F. Werke, Bd. 8. Grundlagen der Geometrie. Teubner, Göttingen, 1900.
2. Pasch, M. Vorlesungen über neuere Geometrie. Teubner, Leipzig, 1882 (2nd ed. by Pasch, M. and Dehn, M., Springer, Berlin, 1926).
3. Hilbert, D. Grundlagen der Geometrie, Festschrift zur Feier der Enthüllung des Gauss-Weber-Denkmals in Göttingen. Teubner, Leipzig, 1899 (Siebente umgearbeitete und vermehrte Auflage, Teubner, Leipzig/Berlin, 1930).
4. Schur, F. Ueber die Grundlagen der Geometrie. Math. Ann., 1901, 55, 265–292.
https://doi.org/10.1007/BF01444974
5. Moore, E. H. On the projective axioms of geometry. Trans. Amer. Math. Soc., 1902, 3, 142–158.
https://doi.org/10.1090/S0002-9947-1902-1500592-8
https://doi.org/10.2307/1986321
6. Veblen, O. A system of axioms for geometry. Trans. Amer. Math. Soc., 1904, 5, 343–384.
https://doi.org/10.1090/S0002-9947-1904-1500678-X
https://doi.org/10.2307/1986462
7. Poincaré, H. Dernières pensées. Ed. Ernest Flammarion, Paris, 1913.
8. Huntington, E. V. and Kline, J. R. Sets of independent postulates for betweenness. Trans. Amer. Math. Soc., 1917, 18, 301–325.
https://doi.org/10.1090/S0002-9947-1917-1501071-5
https://doi.org/10.2307/1988957
9. Huntington, E. V. Set of completely independent postulates for cyclic order. Proc. Nat. Acad. Sci. USA, 1924, 10, 73–78.
https://doi.org/10.1073/pnas.10.2.74
10. Huntington, E. V. A new set of postulates for betweenness with proof of complete independence. Trans. Amer. Math. Soc., 1926, 26, 6–24.
11. Huntington, E. V. The four principal types of order. Trans. Amer. Math. Soc., 1935, 38, 1–9.
https://doi.org/10.1090/S0002-9947-1935-1501800-1
https://doi.org/10.2307/1989726
12. Nuut, J. Topologische Grundlagen des Zahlbegriffs. Acta Comment. Univ. Tartuensis (Dorpatensis), 1929, A15, No. 5.
13. Sarv, J. Geomeetria alused. Acta Comment. Univ. Tartuensis (Dorpatensis), 1931, A19, No. 4.
14. Nuut, J. Einige Bemerkungen über Vierpunktaxiome. Acta Comment. Univ. Tartuensis (Dorpatensis), 1932, A23, No. 4.
15. Tudeberg, A. Über die Beweisbarkeit einiger Anordnungsaussagen in geometrischen Axiomensystemen. Acta Comment. Univ. Tartuensis (Dorpatensis), 1934, A26, No. 6.
16. Lumiste, Ü. Geomeetria alused. Tartu Riiklik Ülikool, Tartu, 1964.
17. Lumiste, Ü. On models of betweenness. Eesti NSV Tead. Akad. Toim. Füüs.-Mat.- ja Tehn.-tead., 1964, 13, 200–209 (in Russian, summaries in Estonian and English; Math. Rev., 1965, 29.6360).
18. Pimenov, R. I. Spaces of kinematic type (a mathematical theory of space-time). Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 1968, 6 (in Russian; English transl.: Kinematic Spaces. Consultation Bureau, New York/London, 1970).
19. Hashimoto, J. Betweenness geometry. Osaka Math. J., 1958, 10, 147–158.
20. Rubinshtein, G. Sh. On an intrinsic characteristic of relatively open convex sets. Dokl. Akad. Nauk SSSR, 1970, 193, 5, 1004–1007 (in Russian).
21. Rubinshtein, G. Sh. Duality in mathematical programming and some problems of convex analysis. Uspekhi Mat. Nauk, 1970, 25, 5(155), 171–201 (in Russian).
https://doi.org/10.1070/RM1970v025n05ABEH003800
22. Rubinshtein, G. Sh. and Rutkovskij, N. V. Axial structures and foundations of geometry. Dokl. Akad. Nauk SSSR, 1980, 254, 6, 1342–1345 (in Russian).
23. Schur, F. Grundlagen der Geometrie. Teubner, Leipzig, 1909.
24. Prenowitz, W. A contemporary approach to classical geometry. Amer. Math. Month., 1961, 68 (No. 1, Part II), 1–67.
25. Prenowitz, W. and Jantosciak, J. Geometries and join spaces. J. reine angew. Math., 1972, 257, 100–128.
https://doi.org/10.1515/crll.1972.257.100
26. Prenowitz, W. and Jordan, M. Basic Concepts of Geometry. Blaisdell, New York, 1965.
27. Bryant, V. W. and Webster, R. J. Generalizations of the theorems of Radon, Helly and Carathéodory. Monatsh. Math., 1969, 73, 309–315.
https://doi.org/10.1007/BF01298981
28. Doignon, J.-P. Caractérisations d'espaces de Pasch-Peano. Acad. Roy. Belg. Bull. Cl. Sci., 1976, 62, 679–699.
https://doi.org/10.3406/barb.1976.58135
29. Bryant, V. W. and Webster, R. J. Convexity spaces. I. J. Math. Anal. Appl., 1972, 37, 206–213.
https://doi.org/10.1016/0022-247X(72)90268-5
30. Bryant, V. W. and Webster, R. J. Convexity spaces. III. J. Math. Anal. Appl., 1977, 57, 382–392.
https://doi.org/10.1016/0022-247X(77)90267-0
31. Cantwell, J. and Kay, D. C. Geometric Convexity. III. Trans. Amer. Math. Soc., 1978, 246, 211–230.
https://doi.org/10.2307/1997972
32. Prenowitz, W. and Jantosciak, J. Join Geometries. A Theory of Convex Sets and Linear Geometry. Springer, New York, 1979.
https://doi.org/10.1007/978-1-4613-9438-9
33. Coppel, W. A. Foundations of Convex Geometry. Cambridge University Press, Cambridge, 1998.
34. Lumiste, Ü. Relationship between join and betweenness geometries. Proc. Estonian Acad. Sci. Phys. Math., 2005, 53, 131–153.
https://doi.org/10.3176/phys.math.2005.3.01
35. Pambuccian, V. A methodologically pure proof of a convex geometry problem. Beiträge zur Algebra und Geometry, 2001, 42, 401–406.
36. Pambuccian, V. Forms of the Pasch axiom in ordered geometry. J. Geometry (submitted).
37. Sörensen, K. Eine Bemerkung zu absoluten Ebenen. Beitr. Algebra Geom., 1993, 24, 23–30.
38. Hall, M. Projective planes. Trans. Amer. Math. Soc., 1943, 54, 229–277.
https://doi.org/10.1090/S0002-9947-1943-0008892-4
39. Skornyakov, L. A. Projective planes. Uspekhi Mat. Nauk, 1951, 6, 6(46), 112–154 (in Russian).
40. Hall, M. The Theory of Groups. MacMillan, 1959 (Russian translation: Moskva, 1962).
41. Moufang, R. Die Einführung der idealen Elemente in die ebene Geometrie mit Hilfe des Satzes vom vollständigen Vierseit. Math. Ann., 1931, 105, 759–778.
https://doi.org/10.1007/BF01455845
42. Sperner, E. Zur Begründung der Geometrie im begrenzten Ebenenstück. Schriften der Königsberger Gelehrten Ges. (Math. Naturwiss. Kl.), 1938, 14. Jahr Heft 6, 121–143; Gesammelte Werke. Heldermann Verlag, Lemgo, 2005, 135–157.
43. Lenz, H. Kleiner Desarguesscher Satz und Dualität in projectiven Ebenen. Jahresb. Deutsch. Math.-Verein., 1954, 57, 381–387.
44. Barlotti, A. Le possibili configurazioni del sistema delle coppie punto-retta (A,a) per cui un piano grafico risulta (A,a)-transitivo. Boll. Un. Mat. Ital., 1957, 12, 212–226.
45. Pickert, G. Projektive Ebenen. Springer, Berlin, 1955 (2nd ed. 1975).
https://doi.org/10.1007/978-3-662-00110-3_2
46. Karzel, H. und Kroll, H.-J. Geschichte der Geometrie seit Hilbert. Wissenschaftliche Buchgesellschaft, Darmstadt, 1988.
47. Priess-Crampe, S. Angeordnete Strukturen: Gruppen, Körper, projektive Ebenen. Springer, Berlin, 1983.
https://doi.org/10.1007/978-3-642-68628-3
48. Jagannathan, T. V. S. Ordered projective planes. J. Indian Math. Soc., 1972, 36, 65–78.
49. Veblen, O. and Wedderburn, J. H. M. Non-Desarguesian and non-Pascalian geometries. Trans. Amer. Math. Soc., 1907, 8, 379–388.
https://doi.org/10.1090/S0002-9947-1907-1500792-1
50. Jousson, J. Konstruktion archimedischer Anordnungen von freien Ebenen. Result. Math., 1981, 4, 55–74.
https://doi.org/10.1007/BF03322966
51. Bruck, R. H. Difference sets in a finite group. Trans. Amer. Math. Soc., 1955, 78, 464–481.
https://doi.org/10.1090/S0002-9947-1955-0069791-3
https://doi.org/10.2307/1993074
52. Skornyakov, L. A. Alternativkörper. Ukrain. Mat. Zhurn., 1950, 2, 70–85 (in Russian).
53. Bruck, R. H. and Kleinfeld, E. The structure of alternative division rings. Proc. Amer. Math. Soc., 1951, 2, 878–890.
https://doi.org/10.1090/S0002-9939-1951-0045099-9
https://doi.org/10.2307/2031702
