\begin{definition}[Definition:Left Naturally Totally Ordered Semigroup]
Let $\struct {S, \circ, \preceq}$ be a positively totally ordered semigroup.
Then $\struct {S, \circ, \preceq}$ is a '''left naturally totally ordered semigroup''' {{iff}}:
:$a \prec b \implies \exists x \in S: b = x \circ a$
\end{definition}