Abstract
The concept nearfield order generalizes the concept field order in a direct manner and was introduced by G. Pickert [14] in form of ordering relations < and by H. Karzel [8] as domains P of positivity. Both were interested in coordinatizing ordered affine and projective planes. W. Kerby [10] was the first to investigate ordered nearfields in detail. He constructed several examples and distinguished between weak and full orders. The interval topologies of nearfield orders were studied by H. Wefelscheid [23]. He remarked that they need not be nearfield topologies. D. Gröger [4] added numerous new results and investigated o-couplings on ordered transcendental field extensions at great length.
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References
Albert, A.A. (1957) A property of ordered rings, Proc. Amer. Math. Soc. 8, 128–129.
Artin, E. and Schreier, O. (1927) Algebraische Konstruktion reeller Körper, Abh. Math. Sem. Univ. Hamburg 5, 85–99.
Fuchs, L. (1961) On the ordering of quotient rings and quotient semigroups, Acta Sci. Math. (Szeged) 22, 42–45.
Gröger, D. (1982) Über angeordnete Fastkörper, Beiträge zur Geometrie und Algebra Techn. Univ. München, 7, Dissertation, Hannover.
Hion, O.B. (1954) Archimedean ordered rings [Russian], Uspechi Mat. Nauk 94, 237–242.
Kalhoff, F. (1988) Eine Kennzeichnung anordnungsfähiger Ternärkörper, J. Geom. 31, 100–113.
Kalhoff, F. (1990) Witt rings of weakly orderable double loops and nearfields, Results Math. 17, 106–119.
Karzel, H. (1955) Ordnungsfunktionen in nichtdesarguesschen projektiven Geometrien, Math. Z. 62, 268–291.
Karzel, H. (1962) Anordnungsfragen in ternären Ringen und allgemeinen projektiven Ebenen, Algebraical and Topological Foundation of Geometry. Proc. Coll. Utrecht 1959, Pergamon Press.
Kerby, W. (1968) Angeordnete Fastkörper, Abh. Math. Sem. Univ. Hamburg 32, 135–146.
Lam, T.Y. (1983) Orderings, valuations and quadratic forms, CBMS Lecture Notes Series. Amer. Math. Soc. Providence, Rhode Islands.
Mathiak, K. (1986) Valuations of skewfields and projective Hjelmslev spaces, Springer, Berlin-Heidelberg-New York.
Neumann, B.H. (1949) On ordered division rings, Trans. Amer. Math. Soc. 66, 202–252.
Pickert, G. (1952) Angeordnete nichtdesarguessche Ebenen, Jber. Deutsch. Math. Verein. 56, 12.
Pickert, G. (1955) Projektive Ebenen, Springer, Berlin-Göttingen-Heidelberg.
Prieß-Crampe, S. (1983) Angeordnete Strukturen: Gruppen, Körper, projektive Ebenen, Springer, Berlin.
Tecklenburg, H. (1992) Stufen der Anordnung in Geometrie und Algebra, Deutscher Universitätsverlag, Wiesbaden.
Wähling, H. (1987) Theorie der Fastkörper, Thales Verlag, Essen.
Wähling, H. (1993) Stetiger Abschluß und Intervalltopologie eines voll angeordneten Fastkörpers, Beiträge zur Geometrie und Algebra 24, 31–35.
Wähling, H. (1994) Approximationssätze für topologische, bewertete und angeordnete Fastkörper, Results Math. 26, 178–194.
Warner, S. (1989) Topological fields, North Holland, Amsterdam.
Weber, H. (1978) Zu einem Problem von H.J. Kowalsky, Abh. Braunschweig. Wiss. Ges. 24, 127–134.
Wefelscheid, H. (1971) Untersuchungen über Fastkörper und Fastbereiche, Habilitationsschrift, Hamburg.
Zariski, O. and Samuel, P. (1960) Commutative Algebra II, Van Nostrand, Toronto-London-Melbourne.
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© 1997 Kluwer Academic Publishers
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Wähling, H. (1997). Ordered Nearfields. In: Saad, G., Thomsen, M.J. (eds) Nearrings, Nearfields and K-Loops. Mathematics and Its Applications, vol 426. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-1481-0_5
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DOI: https://doi.org/10.1007/978-94-009-1481-0_5
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