The Dual of a Chain Geometry

TL;DR

This paper explores the duality of chain geometries, demonstrating that each is canonically isomorphic to its dual and that isomorphisms can be derived from antiisomorphisms of underlying rings.

Contribution

It introduces the concept of the dual of a chain geometry and establishes the canonical isomorphism between a chain geometry and its dual, linking it to ring antiisomorphisms.

Findings

Each chain geometry is canonically isomorphic to its dual.

Isomorphisms of chain geometries can arise from antiisomorphisms of the underlying rings.

The duality concept provides new insights into the structure of chain geometries.

Abstract

We introduce and discuss the dual of a chain geometry. Each chain geometry is canonically isomorphic to its dual. This allows us to show that there are isomorphisms of chain geometries that arise from antiisomorphisms of the underlying rings.