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