Orientation-preserving and orientation-reversing mappings: a new description

TL;DR

This paper characterizes semigroups of mappings that preserve or reverse orientation of finite cycles, linking their actions on triples and quadruples to chord intersection preservation on circles.

Contribution

It introduces a new description of orientation-preserving and reversing mappings through their action on oriented triples and quadruples, connecting to chord intersection preservation.

Findings

Semigroups characterized by actions on oriented triples and quadruples

Equivalence of orientation-reversing semigroup with chord intersection-preserving mappings

Provides a new framework for understanding orientation-preserving/reversing mappings

Abstract

We characterise the respective semigroups of mappings that preserve, or that preserve or reverse orientation of a finite cycle, in terms of their actions on oriented triples and oriented quadruples. This leads to a proof that the latter semigroup coincides with the semigroup of all mappings that preserve intersections of chords on the corresponding circle.