Solutions of the Braid Equation with set-type square

TL;DR

This paper characterizes all non-degenerate solutions to the set-theoretic braid equation on incidence coalgebras derived from height one ordered sets, extending known solutions under specific conditions.

Contribution

It provides a complete classification of solutions on incidence coalgebras extending given solutions on the base set, under certain conditions.

Findings

All non-degenerate solutions extend from base solutions

Solutions are classified for height one ordered sets

Extension conditions are explicitly characterized

Abstract

For a family of height one orders $(X,\le)$ and each non-degenerate solution $r_0\colon X\times X\longrightarrow X\times X$ of the set-theoretic braid equation on $X$ satisfying suitable conditions, we obtain all the non-degenerate solutions of the braid equation on the incidence coalgebra of $(X,\le)$ that extend $r_0$.