Definition:Positively Totally Ordered Semigroup

Definition
Let $\struct {S, \circ, \preceq}$ be a totally ordered semigroup.

Then $\struct {S, \circ, \preceq}$ is a positively totally ordered semigroup :


 * $(1): \forall a, b \in S: a \preceq a \circ b$

and
 * $(2): \forall a, b \in S: b \preceq a \circ b$

Also see

 * Definition:Totally Ordered Semigroup


 * Definition:Naturally Ordered Semigroup
 * Definition:Right Naturally Totally Ordered Semigroup
 * Definition:Left Naturally Totally Ordered Semigroup
 * Definition:Naturally Totally Ordered Semigroup