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Ordered Desarguesian Affine Hjelmslev Planes

Published online by Cambridge University Press:  20 November 2018

L. A. Thomas
L. A. Thomas*
Affiliation:
McMaster University, Hamilton, Ontario
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A Desarguesian affine Hjelmslev plane (D.A.H. plane) may be coordinatized by an affine Hjelmslev ring (A.H. ring), which is a local ring whose radical is equal to the set of two-sided zero divisors and whose principal right ideals are totally ordered (cf. [3]). In his paper on ordered geometries [4], P. Scherk discussed the equivalence of an ordering of a Desarguesian affine plane with an ordering of its coordinatizing division ring. We shall define an ordered D.A.H. plane and follow Scherk's methods to extend his results to D.A.H. planes and their A.H. rings i.e., we shall show that a D.A.H. plane is ordered if and only if its A.H. ring is ordered. We shall also give an example of an ordered A.H. ring. Finally, we shall discuss some infinitesimal aspects of the radical of an ordered A.H. ring.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 21 , Issue 2 , 01 June 1978 , pp. 229 - 235
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Hjelmslev, J., Einleitung in die allgemeine Kongruenzlehre I; KGI. Dansk. Vid. Selsk, Math. Fys. Medd. Mitteilung 8 (1929).Google Scholar
2. Klingenberg, W., Projektive und affine Ebenen mit Nachbarelementen; Math. Z. 60 (1954), 384-406.Google Scholar
3. Lorimer, J. W. and Lane, N. D., Desarguesian affine Hjelmslev planes; J.reine angew. Math. 278/9 (1975), 336-352.Google Scholar
4. Scherk, P., On ordered geometries; Canad. Math. Bull., 6 (1963), 27-36.Google Scholar