Geodesic
Dual Numbers and Topological Hjelmslev Planes
Canadian mathematical bulletin, Tome 26 (1983) no. 3, pp. 297-302

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-1983-048-6
Mots-clés : 51H20, 13J120 
Lorimer, J. W. Dual Numbers and Topological Hjelmslev Planes. Canadian mathematical bulletin, Tome 26 (1983) no. 3, pp. 297-302. doi: 10.4153/CMB-1983-048-6
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[1] 1. Adamson, I. J., Rings, Modules and Algebras, Oliver and Boyd, Edinburgh, 1971. Google Scholar

[2] 2. Artmann, B.. Hjelmslev-Ebenen in projektiven Raumen, Arch. Math. 21 (1970) 304-307. Google Scholar

[3] 3. Artmann, B.. Geometric aspects of primary lattices, Pacific J. Math. 43 (1972) 15-25. Google Scholar

[4] 4. Cronheim, A.. Dual numbers, Witt vectors and Hjelmslev planes, Geom. Dedi. 7 (1978) 287-302. Google Scholar

[5] 5. Dembowski, P.. Finite Geometries, Springer-Verlag (1968). Google Scholar

[6] 6. Engelking, R.. Dimension Theory, North Holland Pub. Co., New York, 1978. Google Scholar

[7] 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] 8. Hewitt, E. and Ross, K. A.. Abstract Harmonic Analysis, Vol. I, Springer-Verlag 1963. Google Scholar

[9] 9. Kaplansky, I.. Notes on ring theory, Mathematical Lecture Notes, University of Chicago, Feb. 1965. Google Scholar

[10] 10. Klingenberg, W.. Projektive und affine Ebenen mit Nachbarelementen, Math. Z 60 (1954), 384-406. Google Scholar

[11] 11. Klingenberg, W.. Desarguessche Ebenen mit Nachbarelementen, Abh. Math. Sem. Univ. Hamburg 20 (1955) 97-111. Google Scholar

[12] 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] 13. Lorimer, J. W.. Topological Hjelmslev planes, Geom. Dedi. 7 (1978) 185-207. Google Scholar

[14] 14. Lorimer, J. W.. Connectedness in topological Hjelmslev planes, Annali di Mat. para ed appl., (iv), Vol. CXVIII (1978) 199-216. Google Scholar

[15] 15. Lorimer, J. W.. Locally compact Hjelmslev planes and rings, Can. J. Math. Vol. XXXIII, No. 4, 1981, 988-1021. Google Scholar

[16] 16. Montgomery, D. and Zippin, L.. Topological Transformation Groups, R. E. Krieger Pub. Co., Huntington, New York, 1974. Google Scholar

[17] 17. Study, E.. Geometrie der Dynamen, Leipzig, 1903. Google Scholar

[18] 18. Türner, G.. Eine Klassifizierung von Hjelmslev-ring und Hjelmslev-Ebenen, Mitt. Math. Sem. Giessen 107 (1974). Google Scholar

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