Solutions of the Braid Equation with set-type square

For a family of height one orders $(X,le)$ and each non-degenerate solution $r_0colon Xtimes Xlongrightarrow Xtimes X$ 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 $r_0$.