Positive existential definability of parallelism in terms of betweenness in Archimedean ordered affine geometry
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 \textit{X.-D. Hou} and \textit{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.
scientific article; zbMATH DE number 5959795
scientific article