Let $\mathbf{X} = (X,+,\leq)$ be a partially ordered magma, i.e. $(X,+)$ is a magma (written additively), and $\leq$ is a partial order on $X$ satisfying: for every $x,y,z \in X$ such that $x \leq y$, $x+z \leq y+z$ and $z+x \leq z+y$.
Is there a name for the following property of $\mathbf{X}$, which is reminiscent of the property defining a Euclidean domain?
For every $a, b \in X$ there exist $q, r \in X$ such that $a = q + r$ and $r \leq b$. Moreover, the set of such $r$'s has a maximal element.
If there's a name for the property described in just the first of these two sentences, I'll be interested to know it too.