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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

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