A new formulation of the semigroup of orientation-preserving and orientation-reversing mappings

TL;DR

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

Contribution

It introduces a new formulation connecting orientation-preserving/reversing mappings with chord intersection preservation, providing a unified characterization.

Findings

Semigroups characterized by actions on oriented triples and quadruples

Equivalence of orientation-preserving/reversing semigroups with chord intersection preserving mappings

New proof linking geometric and algebraic properties of mappings

Abstract

We characterize 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.