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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.

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:

Zbl 1167.26004
Full Text: DOI

References:

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