On ordered left groups

Our purpose is to give some similarities and some differences concerning the left groups between semigroups and ordered semigroups. Unlike in semigroups (without order) if an ordered semigroup is left simple and right cancellative, then it is not isomorphic to a direct product of a zero ordered semigroup and an ordered group, in general. Unlike in semigroups (without order) if an ordered semigroup $S$ is regular and has the property $aS\subseteq (Sa]$ for all $a\in S$, then the $\mathcal N$-classes of $S$ are not left simple and right cancellative, in general. The converse of the above two statements hold both in semigroups and in ordered semigroups. Exactly as in semigroups (without order), an ordered semigroup is a left group if and only if it is regular and right cancellative.