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Fastgeordnete affine lokale Ternärringe und pregeordnete Biternärringe. (Quasiordered affine local ternary rings and preordered biternary rings.). (German) Zbl 0711.51001

In Czech. Math. J. 30(105), 556-568 (1980; Zbl 0467.51016), the author has codified the notion of preordered affine local ternary rings (fastgeordnete affine lokale Ternärringe) as an algebraic counterpart of the concept of arbitrary preordered affine Klingenberg planes.
For the special case of affine Hjelmslev planes, in J. Geom. 23, 14-44 (1984; Zbl 0545.51006)], C. A. Baker, N. D. Lane, J. W. Lorimer, and J. A. A. Laxton have introduced another notion of preorderings, geometrically equivalent to the author’s, but related to a quite different algebraic setting in terms of preordered Hjelmslev biternary rings. (This paper seems not to be correctly quoted by the author; [3] 4.1 should read [3] 5.1 and [3] Lemma 7, Beh. 6 should be [3] 5.4.)
The author gives an algebraic proof that, in case the underlying plane is Hjelmslev, both algebraic concepts are equivalent.
Reviewer: F.Kalhoff

MSC:

51G05 Ordered geometries (ordered incidence structures, etc.)
51C05 Ring geometry (Hjelmslev, Barbilian, etc.)
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References:

[1] Bacon P.Y.: Coordinatized Hjelmslev planes. Ph.D. thesis, University of Florida, Gainesville, Florida, 1974.
[2] Bacon P.Y.: An Introduction to Klingenberg Planes. Volume 1, Publised by P.Y.Bacon, 1976. · Zbl 0372.50003
[3] Baker C.A., Lane N.D., Laxton J.A.A., Lorimer J.W.: Ordered Affine Hjelmslev Planes. Archiv der Mathematik. · Zbl 0608.51007
[4] Machala F.: Koordinatisation affiner Ebenen mit Homomorphismus. Math.Slovaca, 27, 1977, No.2, 181-193. · Zbl 0359.50017
[5] Machala F.: Biternärringe und affine lokale Ternärringe. Acta Univ.Pal. · Zbl 0693.51002
[6] Machala F.: Angeordnete affine lokale Ternärringe und angeordnete affine Klingenbergsche Ebenen. Czech.Math. Journal, 30 (105), 1980, 556-568. · Zbl 0467.51016
[7] Machala F.: Fastgeordnete und geordnete affine Klingenbergsche Ebenen. Čas.pro p\?st.mat. 106, 1981, 138-155. · Zbl 0475.51003
[8] Machala F.: Fastgeordnete und geordnete lokale Ringe und ihre geometrische Anwendung. Čas.pro p\?st.mat. 106, 1981, 269-278. · Zbl 0476.51004
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