In this question, it is shown that all Archimedean ordered groups are isomorphic to an ordered subgroup of $\mathbb R$. Additionally, it is shown that if such a group is complete, then it is isomorphic to the trivial group, $\mathbb Z$, or $\mathbb R$.

I'm curious if a similar result holds it we loosen the group condition to being a monoid. In particular,

- Are all Archimedean linearly ordered monoids isomorphic to ordered submonoids of $\mathbb R$?
- If the order is complete, must it be isomorphic to $\mathbb R^{\ge 0}$, $\mathbb R$, or some ordered submonoid of $\mathbb Z$?

If not, are there additional properties like cancellation or commutativity that could be assumed to make these true?