\begin{definition}[Definition:Positively Totally Ordered Semigroup]
Let $\struct {S, \circ, \preceq}$ be a totally ordered semigroup.
Then $\struct {S, \circ, \preceq}$ is a '''positively totally ordered semigroup''' {{iff}}:
:$(1): \forall a, b \in S: a \preceq a \circ b$
and
:$(2): \forall a, b \in S: b \preceq a \circ b$
\end{definition}