Definition:Left Naturally Totally Ordered Semigroup

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

Then $\struct {S, \circ, \preceq}$ is a left naturally totally ordered semigroup for all $a, b \in S$:


 * $a < b$ implies that for some $y \in S$, $b = x \circ a$.

Also see

 * Definition:Totally Ordered Semigroup


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