The Green's relations and the generalized Green's relations on certain transformation semigroups

We investigate the Green's relations $\mathscr L$, $\mathscr R$ on the semigroups of isotone transformations of the partially ordered sets, and also the generalized Green's relations $\mathscr L^*$, $\mathscr R^*$ on the semigroup $B(X)$ of binary relations on a set $X$. It is proved that $\mathscr L=\mathscr L^*$, $\mathscr R=\mathscr R^*$ in the semigroup $B(X)$ though this semigroup is non-regular for $|X|\ge3$.