Definition:Left Naturally Totally Ordered Semigroup

Definition
Let $\left({S, \circ, \preceq}\right)$ be a positively totally ordered semigroup.

Then $\left({S, \circ, \preceq}\right)$ is a left naturally totally ordered semigroup iff for all $a, b \in S$:


 * $a < b$ implies that for some $y \in S$, $b = x \circ a$.