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Axioms for Absolute Geometry. III

Published online by Cambridge University Press:  20 November 2018

J. F. Rigby
J. F. Rigby*
Affiliation:
University College, Cardiff, Wales
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This paper is a continuation of [1; 2]. In [2], I stated that I had been unable to construct examples of planes satisfying various conditions. Some of the examples that I have since constructed are given below. A discussion of one-dimensional absolute geometries, with examples, will be given in a separate paper. The relevant parts of [1] and [2] are [1, § 1, § 2 up to 2.4; 2, § 2]. We shall use the notation and terminology of [1; 2]; the axioms Cl*-C4* and C4** (referred to below) can all be found in [1].

We shall show here that spaces of dimension greater than 1 exist, both Archimedean and non-Archimedean, satisfying Cl*-C4*, in which not all points are isometric, and that C4** does not follow from Cl*-C4* in non- Archimedean geometries of dimension greater than 1.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 1 , 01 February 1970 , pp. 185 - 190
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Rigby, J. F., Axioms for absolute geometry, Can. J. Math. 20 (1968), 158–181.Google Scholar
2. Rigby, J. F., Axioms for absolute geometry. II, Can. J. Math. 21 (1969), 876–883.Google Scholar
3. Sierpinski, W., Cardinal and ordinal numbers, Polska Akademia Nauk, Monographie Matematyczne, Tom 34 (Panstwowe Wydawnictwo Naukowe, Warsaw, 1958).Google Scholar