I've found myself looking at a structure $\mathbb{M}$ whose important properties are:

1. $\mathbb{M}$ is a discretely ordered additive monoid.
2. $\mathbb{M}$ has a least element, and this least element is the additive identity $0\in\mathbb{M}$.
3. All elements $m\in\mathbb{M}$ have unique additive decompositions once we choose some $n<m$ in $\mathbb{M}$ as one additive factor of $m$. That is, for all $n\in\mathbb{M}$ such that $n<m$, there exists exactly one $l\in\mathbb{M}$ such that $n+l=m$.

Does this structure have a name\has it been studied at all? A simple example is the natural numbers $\mathbb{N}$ under standard addition and ordering, another example is the non-negative part of the ordered Grothendieck group $\mathfrak{G}^+(\omega^\alpha)$ of a $\gamma$-number $\omega^\alpha$ under natural addition and standard ordinal ordering for $\alpha\in O_n$ fixed, or a proper-class sized example is $\mathfrak{G}^+(O_n).$