Definition 2

A Hall set relative to $X$ is any totally ordered subset $H$ of $M(X)$ satisfying the following conditions: (A) If $u \in H, v \in H$ and $l(u) < l(v)$, then $u < v$. (B) $X \subset H$ and $H \cap M^2(X)$ consists of the products $xy$ with $x, y$ in $X$ and $x < y$. (C) An element $w$ of $M(X)$ of length $\geqslant 3$ belongs to $H$ if and only if it is of the form $a(bc)$ with $a, b, c$ in $H$, $bc \in H$, $b \leqslant a < bc$ and $b < c$.