Solutions of the Braid Equation with set-type square

For a family of height one orders $(X,\le )$ and each non-degenerate solution $r_0\text{\hspace{0.17em}⁣}:X\times X⟶X\times X$ 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 $r_0$.