Generating geometry axioms from poset axioms

TL;DR

This paper explores how fundamental order axioms like transitivity and antisymmetry, when viewed from intervals instead of points, generate well-known axioms of order geometry, linking classical geometric principles.

Contribution

It introduces a novel perspective by deriving order geometry axioms from poset axioms through interval-based viewpoints, connecting classical and modern geometric axioms.

Findings

Transitivity from an interval is equivalent to Peano's Assioma XIII.

Antisymmetry from an interval is equivalent to Pasch's VIII.

Several known implications between axioms are proved.

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.