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Dual Numbers and Topological Hjelmslev Planes

Published online by Cambridge University Press:  20 November 2018

J. W. Lorimer
J. W. Lorimer*
Affiliation:
Department of Mathematics, University of TorontoToronto, Canada, M5S1A1
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Abstract

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In 1929 J. Hjelmslev introduced a geometry over the dual numbers ℝ+tℝ with t2 = Q. The dual numbers form a Hjelmslev ring, that is a local ring whose (unique) maximal ideal is equal to the set of 2 sided zero divisors and whose ideals are totally ordered by inclusion. This paper first shows that if we endow the dual numbers with the product topology of ℝ2, then we obtain the only locally compact connected hausdorfT topological Hjelmslev ring of topological dimension two. From this fact we establish that Hjelmslev's original geometry, suitably topologized, is the only locally compact connected hausdorfr topological desarguesian projective Hjelmslev plane to topological dimension four.

Keywords

51H2013J120
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 26 , Issue 3 , 01 September 1983 , pp. 297 - 302
Copyright
Copyright © Canadian Mathematical Society 1983

References

1. Adamson, I. J., Rings, Modules and Algebras, Oliver and Boyd, Edinburgh, 1971.Google Scholar
2. Artmann, B.. Hjelmslev-Ebenen in projektiven Raumen, Arch. Math. 21 (1970) 304-307.Google Scholar
3. Artmann, B.. Geometric aspects of primary lattices, Pacific J. Math. 43 (1972) 15-25.Google Scholar
4. Cronheim, A.. Dual numbers, Witt vectors and Hjelmslev planes, Geom. Dedi. 7 (1978) 287-302.Google Scholar
5. Dembowski, P.. Finite Geometries, Springer-Verlag (1968).Google Scholar
6. Engelking, R.. Dimension Theory, North Holland Pub. Co., New York, 1978.Google Scholar
7. Hjelmslev, J.. Einleitung in die allgemeine Kongruenzlehre Danske Vid. Selsk Mat. Fys. Medd. 8 nr. 11 (1929); 10 nr. 1 (1929); 19 nr. 12 (1942); 22 nr. 6 and nr. 13 (1945); 25, nr. 10 (1949).Google Scholar
8. Hewitt, E. and Ross, K. A.. Abstract Harmonic Analysis, Vol. I, Springer-Verlag 1963.Google Scholar
9. Kaplansky, I.. Notes on ring theory, Mathematical Lecture Notes, University of Chicago, Feb. 1965.Google Scholar
10. Klingenberg, W.. Projektive und affine Ebenen mit Nachbarelementen, Math. Z 60 (1954), 384-406.Google Scholar
11. Klingenberg, W.. Desarguessche Ebenen mit Nachbarelementen, Abh. Math. Sem. Univ. Hamburg 20 (1955) 97-111.Google Scholar
12. Lorimer, J. W. and Lane, N. D.. Desarguesian affine Hjelmslev planes, Journal für die reine und angewandte Mat. Band 278/279 (1975) 336-352.Google Scholar
13. Lorimer, J. W.. Topological Hjelmslev planes, Geom. Dedi. 7 (1978) 185-207.Google Scholar
14. Lorimer, J. W.. Connectedness in topological Hjelmslev planes, Annali di Mat. para ed appl., (iv), Vol. CXVIII (1978) 199-216.Google Scholar
15. Lorimer, J. W.. Locally compact Hjelmslev planes and rings, Can. J. Math. Vol. XXXIII, No. 4, 1981, 988-1021.Google Scholar
16. Montgomery, D. and Zippin, L.. Topological Transformation Groups, R. E. Krieger Pub. Co., Huntington, New York, 1974.Google Scholar
17. Study, E.. Geometrie der Dynamen, Leipzig, 1903.Google Scholar
18. Türner, G.. Eine Klassifizierung von Hjelmslev-ring und Hjelmslev-Ebenen, Mitt. Math. Sem. Giessen 107 (1974).Google Scholar