Positive existential definability of parallelism in terms of betweenness in Archimedean ordered affine geometry. (English) Zbl 1230.51013
The authors show that in Archimedean ordered affine planes, parallelism is positively existentially definable in terms of betweenness and negation of equality. This can be used to prove the following result of X.-D. Hou and G. McColm [Rocky Mt. J. Math. 38, No. 1, 123–137 (2008; Zbl 1167.26004)]: every self-map of an Archimedean ordered pappian affine plane preserving betweenness in both directions is surjective.
Reviewer: Theo Grundhöfer (Würzburg)
MSC:
| 51G05 | Ordered geometries (ordered incidence structures, etc.) |
| 03C65 | Models of other mathematical theories |
| 51F20 | Congruence and orthogonality in metric geometry |
| 51F05 | Absolute planes in metric geometry |
Citations:
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