Semigroup $(S,*)$ is naturally ordered (natural ordering is supposed to behave like natural numbers under addition) iff:
$1$. ${\forall}I\,{\subseteq}\,S:{\exists}m\,{\in}\,I:{\forall}n\,{\in}\,I:m\,{\preceq}\,n$
$2$. ${\forall}a,b,c\,{\in}\,S:a*b=a*c{\implies}b=c$
$3$. ${\forall}d,e\,{\in}\,S:d\,{\preceq}\,e:{\exists}f\,{\in}\,S:f*d=e$
$4$. ${\exists}p,r\,{\in}\,S:p≠r$
All naturally ordered semigroups are commutative. This apparently follows from the axioms. How?