Generating geometry axioms from poset axioms

Abstract

Two axioms of order geoemtry are the poset axioms of transitivity and antisymmetry of the relation "is in front of" when looking from a point. From these axioms, by looking from an interval instead of a point, further well-known axioms of order geometry are generated in the following sense: Transitivity when looking from an interval is equivalent to Assioma XIII of paragraph 10 in G. Peano, I principii di geometria logacimente exposti. Assuming this axiom, antisymmetry when looking from an interval is equivalent VIII. Grundsatz in paragraph 1 in M. Pasch, Vorlesungen ueber neuere Geometrie. Further equivalences, with some of the implications well-known, are proved along the way.