A common axiomatic basis for projective geometry and order geometry

TL;DR

This paper establishes a unified axiomatic framework linking projective geometry and order geometry, demonstrating their deep connection through a common set of axioms and concepts.

Contribution

It presents a one-to-one correspondence between projective spaces and join spaces, introducing a new projectivity criterion and linking these geometries to matroid theory.

Findings

Established a correspondence between projective and join spaces.

Proved a new projectivity criterion for join spaces.

Connected projective join spaces to matroid theory.

Abstract

A natural one-to-one correspondence between projective spaces, defined by an axiom system published by O. Veblen and J. W. Young in 1908, and projective join spaces, defined by an axiom system published by M. Pieri in 1899, is presented. A projecitivity criterion for join spaces is proved that amounts to replacing one of the projective geometry axioms of Pieri by an axiom published by G. Peano in 1889 as part of an axiom system for order geometry. Thus, projective geometry and order geometry have a broad common axiomatic basis. As a corollary, it is shown how the concept of a projective join space can be derived from the concept of a matroid. The defining properties of an equivalence relation are used as a conceptual red thread.