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 $\mathfrak{G}^+(O\_n).$



Name for this algebraic structure?
As Emil Jeřábek correctly suggests in the comments above, $\mathbb{M}$ is exactly the non-negative part of a discretely ordered group -- I humbly offer a proof to close the thread. The proof that the non-negative part of a discretely ordered group satisfies the above requirements is trivial.


For the opposite direction, we prove that $\mathbb{M}$ uniquely determines (up to isomorphism) a discretely ordered group whose set of non-negative elements is isomorphic to $\mathbb{M}$ as a discretely ordered monoid. Let $+$ and $\leq$ denote the addition and ordering in $\mathbb{M}$, and consider $\mathbb{M}\times\mathbb{M}$ with the binary operation $+\_{\_\mathbb{M}}:(\mathbb{M}\times\mathbb{M})\times(\mathbb{M}\times\mathbb{M})\rightarrow\mathbb{M}\times\mathbb{M}$ and the unary operation $-\_{\_\mathbb{M}}:\mathbb{M}\times\mathbb{M}\rightarrow\mathbb{M}\times\mathbb{M}$ given by: $$(m\_0,m\_1)+\_{\_\mathbb{M}}(m\_2,m\_3)=(m\_0+m\_2,m\_1+m\_3),$$ $$-\_{\_\mathbb{M}}(m\_0,m\_1)=(m\_1,m\_0),$$ and additive identity $0\_{\_\mathbb{M}}=(0,0)$, together with the ordering $\leq\_{\_\mathbb{M}}\subseteq\mathbb{M}^4$ given by: $$(m\_0,m\_1)\leq\_{\_\mathbb{M}}(m\_2,m\_3)\iff m\_0+m\_3\leq m\_1+m\_2.$$ We then mod out by the congruence relation $\equiv\subseteq\mathbb{M}^4$ defined by $$(m\_0,m\_1)\equiv(m\_2,m\_3)\iff m\_0+m\_3=m\_1+m\_2$$ to handle the fact that we want to forget the multiple coordinate representations of a given group element. We now have that $\mathbb{G}\_{\_\mathbb{M}}=\big\langle(\mathbb{M}\times\mathbb{M})\setminus\equiv,+\_{\_\mathbb{M}},-\_{\_\mathbb{M}},\leq\_{\_\mathbb{M}},0\_{\_\mathbb{M}}\big\rangle$ is a discretely ordered group, and $$\mathbb{G}\_{\_\mathbb{M}}^+=\{(m,0):m\in\mathbb{M}\}$$ is isomorphic to $\mathbb{M}$ as a discretely ordered monoid under the first factor projection $\pi\_0:\mathbb{M}\times\mathbb{M}\rightarrow\mathbb{M}$. We could easily treat the first factor as being negative instead of positive by tweaking the definition of $\leq\_{\_\mathbb{M}}$, however the construction would still obviously be isomorphic to $\mathbb{G}\_{\_\mathbb{M}}$.


