Definition:Positively Totally Ordered Semigroup

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

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


 * $a \preceq a \circ b$
 * $b \preceq a \circ b$