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Moulton Affine Hjelmslev Planes

Published online by Cambridge University Press:  20 November 2018

Catharine Baker
Catharine Baker*
Affiliation:
Dept. of Mathematics, McMaster University, Hamilton, Ontario
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Desarguesian affine Hjelmslev planes (D.A.H. planes) were introduced by Klingenberg in [1] and generalized by Lane and Lorimer in [2]. D.A.H. planes are coordinatized by affine Hjelmslev rings (A.H. rings) which are local rings whose radicals are equal to their sets of two-sided zero divisors and whose principal right ideals are totally ordered. In [5], ordered D.A.H. planes were defined and the induced orderings of their A.H. rings were discussed. In this note an ordered non-Desarguesian A.H. plane is constructed from an arbitrary ordered D.A.H. plane. The existence of such planes ensures that the discussion of ordered non-Desarguesian A.H. planes by J. Laxton in [3] is meaningful. The basic idea employed is essentially the same as the one used in the construction of the classical Moulton plane from the real affine plane (cf. [4])

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 21 , Issue 2 , 01 June 1978 , pp. 135 - 142
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Klingenberg, W.: Desarguessche Ebenen mit Nachbarelementen. Abh. Math. Sem. Univ. Hamburg 20 (1955), 97-111.Google Scholar
2. Lane, N. D. and Lorimer, J. W.: Desarguesian affine Hjelmslev planes. J. reine angew. Math. 278/279 (1975), 336-352.Google Scholar
3. Laxton, J. A. A.: Ordered non-Desarguesian Affine Hjelmslev Planes. M.Sc. thesis, McMaster University, Hamilton (1976).Google Scholar
4. Moulton, F. R.: A simple non-Desarguesian plane geometry. Trans. Amer. Math. Soc. 3 (1902), 192-195.Google Scholar
5. Thomas, L. A.: Ordered Desarguesian affine Hjelmslev-planes. M.Sc. thesis, McMaster University, Hamilton (1975).Google Scholar