Exercise 3, § 2

Let $X$ be a set with a total ordering. Show that there exists a Hall set relative to $X$ such that $H \cap M^3(X)$ consists of the products $z(yx)$ with $y < x$, $y \leqslant z$ and that $H \cap M^4(X)$ consists of the products $w(z(yx))$ with $w \geqslant z \geqslant y$, $y < x$ and the products $(ab)(cd)$ with $a < b, c < d$ and either $a < c$ or $a = c$ and $b < d$.