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Locally Compact Hjelmslev Planes and Rings

Published online by Cambridge University Press:  20 November 2018

J. W. Lorimer
J. W. Lorimer*
Affiliation:
University of Toronto, Toronto, Ontario
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Affine and projective Hjelmslev planes are generalizations of ordinary affine and projective planes where two points (lines) may be joined by (may intersect in) more than one line (point). The elements involved in multiple joinings or intersections are neighbours, and the neighbour relations on points respectively lines are equivalence relations whose quotient spaces define an ordinary affine or projective plane called the canonical image of the Hjelmslev plane. An affine or projective Hjelmslev plane is a topological plane (briefly a TH-plane and specifically a TAH-plane respectively a TPH-plane) if its point and line sets are topological spaces so that the joining of non-neighbouring points, the intersection of non-neighbouring lines and (in the affine case) parallelism are continuous maps, and the neighbour relations are closed.

In this paper we continue our investigation of such planes initiated by the author in [38] and [39].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 4 , 01 August 1981 , pp. 988 - 1021
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Adamson, I. J., Rings, modules and algebras (Oliver and Boyd, Edinburgh, 1971).Google Scholar
2. Armacost, D. L. and Armacost, W. L., Uniqueness in structure theorems for L.C.A. groups, Can. J. Math. 30 (1978), 593–599.Google Scholar
3. Artmann, B., Desarguessche Hjelmslev-Ebenen n-ter, Stufe. Mitt. Math. Sem. GieBen 91 (1971), 1–19.Google Scholar
4. Artmann, B., Geometric aspects of primary lattices, Pacific J. Math. 1$ (1972), 15–25.Google Scholar
5. Artmann, B., Hjelmslev-Ebenen in projektiven Raumen, Arch. Math. 21 (1970), 304–307.Google Scholar
6. Baker, C., Moulton affine Hjelmslev planes, Can. Math. Bull. 21 (1978).Google Scholar
7. Bing, R. H. and Borsuk, K., Some remarks concerning topological homogeneous spaces, Ann. Math. 81 (1965), 100–111.Google Scholar
8. Bourbaki, N., General topology, Vol. I, II (Addison-Wesley Pub. Co., Reading, Mass., 1966).Google Scholar
9. Bourbaki, N., Commutative algebra (Addison-Wesley Pub. Co., Reading, Mass., 1972).Google Scholar
10. Brungs, H. H. and Törner, G., Chain rings and prime ideals, Archiv Der Math. 27 (1976), 253–260.Google Scholar
11. Cronheim, A., Dual numbers, Witt vectors, and Hjelmslev planes, Geometric Dedicata 7 (1978), 287–302.Google Scholar
12. Clark, E. W. and Drake, D. A., Finite chain rings, Abh. Math. Sem. Univ. Hamburg 39 (1973), 147–153.Google Scholar
13. Drake, D. A., On n-uniform Hjelmslev planes, Journal of Combinatorial Theory 9 (1970), 267–288.Google Scholar
14. Drake, D. A., Existence of parallelism and projective extensions for strongly n-uniform near affine Hjelmslev planes, Geom. Ded. 3 (1974), 295–324.Google Scholar
15. Dugundji, J., Topology (Allyn and Bacon Inc., 1966).Google Scholar
16. Ellis, R., A note on the continuity of the inverse, Proc. Amer. Math. Soc. 8 (1957), 372–373.Google Scholar
17. Engelking, R., Outline of general topology (John Wiley and Sons Inc., New York, 1968).Google Scholar
18. Engelking, R., Dimension theory (North Holland Pub. Co., New York, 1978).Google Scholar
19. Fâry, L., Dimension of the square of a space, Bull. Amer. Math. Soc. 67 (1961), 135–137.Google Scholar
20. Goldman, O. and Sah, C. H., Locally compact rings of special type, Journal of Algebra 11 (1969), 363–454.Google Scholar
21. Hewitt, E. and Ross, K. A., Abstract harmonic analysis, Vol. I (Springer-Verlag, 1963).Google Scholar
22. Honking, J. G. and Young, G. S., Topology (Addison-Wesley Pub. Co., 1961).Google Scholar
23. Hurewitzand, W., Wallman, H., Dimension theory (Princeton University Press, 1941).Google Scholar
24. Husain, T., Introduction to topological groups (W. B. Saunders Co., 1966).Google Scholar
25. Jacobson, N. and Taussky, O., Locally compact rings, Proc. Nat. Acad. Sci., U.S.A. 21 (1935), 106–108.Google Scholar
26. Jacobson, N., Theory of rings, Math. Surveys 11, Amer. Math. Soc. (1943).Google Scholar
27. Kaplansky, I., Topological rings, American Journal of Math. 69 (1947), 153–183.Google Scholar
28. Kaplansky, I., Locally compact rings, American Journal of Math. 70 (1948), 447–459.Google Scholar
29. Kaplansky, I., Locally compact rings II, Amer. Journal of Math. 73 (1951), 20–24.Google Scholar
30. Kaplansky, I., Locally compact rings III, Amer. Journal of Math. 74 (1952), 929–935.Google Scholar
31. Kaplansky, I., Topological methods in valuations theory, Duke Math. J. 14 (1947), 527–541.Google Scholar
32. Kaplansky, I., Dual rings, Annals of Math. 49 (1948), 689–701.Google Scholar
33. Klingenberg, K., Projektive und affine Ebenen mit Nachbarelementen, Math. Z. 60 (1954), 384–406.Google Scholar
34. Klingenberg, K., Desargues s che Ebenen mit Nacharelementen, Abh. Math. Sem. Univ. Hamburg 20 (1955), 97–111.Google Scholar
35. Lambek, J., Lectures on rings and modules (Blaisdell Pub. Co., 1966).Google Scholar
36. Lorimer, J. W. and Lane, N. D., Desarguesian affine Hjelmslev planes, Journal fur die reine und angewandte Mat. Band 278/279, (1975), 336–352.Google Scholar
37. Lorimer, J. W., Coordinate theorems for affine Hjelmslev planes, Ann. Math. Pura Appl. 105 (1975), 171–190.Google Scholar
38. Lorimer, J. W., Topological Hjelmslev planes, Geom. Dedicata 7 (1978), 185–207.Google Scholar
39. Lorimer, J. W., Connectedness in topological Hjelmslev planes, Annali di Mat. pura ed appl. 118 (1978), 199–216.Google Scholar
40. Löwen, R., Vierdimensionale Stabile Ebenen, Geom. Dedi. 5 (1976), 239–294.Google Scholar
41. Lùneburg, H., Affine Hjelmslev-Ebenen mit transitiver Translationgruppe, Math. Z. 79 (1962), 260–283.Google Scholar
42. Massey, W. S., Algebraic topology: An introduction, Graduate Texts in Mathematics 56 (Springer-Verlag, New York, 1967).Google Scholar
43. Montgomery, D. and Zippin, L., Topological transformation groups (R. E. Krieger Pub. Co., Huntington, New York, 1974).Google Scholar
44. Pickert, G., Projektive Ebenen (Springer-Verlag, 1975).Google Scholar
45. Pontrjagin, L., Topological groups (Princeton University Press, 1939).Google Scholar
46. Salzmann, H., Topologische projektive Ebenen, Math. Z. 67 (1957), 436–466.Google Scholar
47. Salzmann, H., Über der Zusammenhang in topologischen projectiven Ebenen, Math. Z. 61 (1955), 489–494.Google Scholar
48. Salzmann, H., Topological planes, Advances in Mathematics, 2, Fascicle 1 (1967).Google Scholar
49. Stroyan, K. D. and Luxemburg, W. A. J., Introduction to the theory of infinitesimals (Academic Press, New York, 1976).Google Scholar
50. Thomas, L. A., Ordered desarguesian affine Hjelmslev planes, Can. Math. Bull. 21 (1978), 229–235.Google Scholar
51. Törner, G., Eine Klassifizierung von Hjelmslev-ring und Hjelmslev-Ebenen, Mitt. Math. Sem. Giessen 107 (1974).Google Scholar
52. Törner, G., Über den Stufenaufbau von Hjelmslev-Ebenen, Mitt. Math. Sem. Giessen 126 (1977).Google Scholar
53. Törner, G., Hjelmslev-Ringe und Géométrie der Nachbarschaftsbereiche in den zugehörigen Hjelmslev-Ebenen (Giessen, Diplomarbeit, 1972).Google Scholar
54. van Kampen, E. R., Locally compact abelian groups, Proc. Nat. Acad. Sci. U.S.A. 20 (1934), 434–436.Google Scholar
55. Weil, A., Vintegration dans les groupes topologiques et ses applications (Paris, Herman, 1940).Google Scholar
56. Whyburn, G. T., Analytic topology, Amer. Math. Soc. Coll. Pub. 28 (1942).Google Scholar