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Keywords:

  • Quantifier elimination;
  • euclidean geometry;
  • affine geometry;
  • geometric constructions;
  • semi-algebraic geometry;
  • MSC (2010) 03C10;
  • 51M05;
  • 03B30

Abstract

We introduce new first-order languages for the elementary n-dimensional geometry and elementary n-dimensional affine geometry (n ≥ 2), based on extending equation image and equation image, respectively, with new function symbols. Here, β stands for the betweenness relation and ≡ for the congruence relation. We show that the associated theories admit effective quantifier elimination.