Nearaffine planes related to pseudo-ordered fields

• Autorzy: Jakóbowski J.
• Języki publikacji: EN

Abstrakty

EN

The constructions of affine planes and Minkowski planes related to pseudo-ordered fields are given in [3] and [2], respectively. We here give some analogous construction for nearaffine planes. Like before, we shall use some functions f, g and determine some conditions on f, g, necessary and suffcient to get the required plane. The Veblen postulate has a particular meaning in nearaffine planes, so it is also considered in the work. Some special cases like the field of the reals and finite fields of odd order are investigated, too. We give some examples of such nearaffine planes and consider their particular automorphisms. Every Minkowski plane related to pseudo-ordered field F determines a nearaffine plane connected with F [2, Proposition 1, p. 187]. But only weaker version of the reciprocal statement is true, i.e. a nearaffine plane related to a pseudo-ordered field determines a hyperbola structure (i.e. Minkowski plane without touching axiom).

Słowa kluczowe

EN

automorphism; Minkowski planes; nearaffine planes; pseudo-ordered fields

PL

automorfizm; płaszczyzna Minkowskiego; pseudo-uporządkowane ciało

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2002
• Tom: Vol. 50, no 3
• Strony: 345--360
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Jakóbowski J.

• University of Warmia and Mazury in Olsztyn, Faculty of Mathematics and Informatics, ul. Żołnierska 14, 10-561 Olsztyn, Poland,

Bibliografia

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• [2] J. Jakóbowski, A new construction for Minkowski planes, Geom. Dedicata, 69 (1968) 179-188.
• [3] J. Jakóbowski, A new generalization of Moulton affine planes, Geom. Dedicata, 42 (1992) 243-253.
• [4] J. Jakóbowski, Affine planes and Minkowski planes induced by pseudo-ordered fields (in Polish), National Conference of Applications of Statistics, ART Olsztyn (1996) 58-69.
• [5] J. Jakóbowski, A. Martaś, Multicenral automorphisms in geometries of circles, Bull. Pol. Ac.: Math., 49 (4) (2001) 417-432.
• [6] M. Klein, H. J. Kroll, A classification of Minkowski planes, J. Geom., 36 (1989) 99-109.
• [7] F. R. Moulton, A simple non-Desarguesian plane geometry, Trans. Amer. Math. Soc., 3 (1902) 192-195.
• [8] N. Percsy, Finite Minkowski planes in which every circle-symmetry is an automorphism, Geom. Dedicata, 10 (1981) 269-282.
• [9] W. A. Pierce, Moulton planes, Can. J. Math., 13 (1961) 427-436.
• [10] H. A. Wilbrink, Finite Minkowski planes, Geom. Dedicata, 12 (1982) 119-129.
• [11] H. A. Wilbrink, Nearaffine planes, Geom. Dedicata, 12 (1982) 53-62.