Positive existential definability of parallelism in terms of betweenness in archimedean ordered affine geometry

Franz Kalhoff, Victor Pambuccian

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Abstract

We prove that one can define the relation ∥, with ab ∥ cd to be read as 'a = 6 or c = d or aft and cd are parallel lines (or coincide)' positively existentially in Lw1win terms of ≠ and the ternary relation B of betweenness, with B(abc) to be read as '6 lies between a and c' in Archimedean ordered affine geometry. We also show that a self-map of an Archimedean ordered translation plane or of a flat affine plane which preserves both B and ¬B must be a surjective affine mapping.

Original languageEnglish (US)
Pages (from-to)1501-1521
Number of pages21
JournalRocky Mountain Journal of Mathematics
Volume41
Issue number5
DOIs
StatePublished - Nov 25 2011

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ASJC Scopus subject areas

  • Mathematics(all)

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