Let $A$ be a commutative monoid and define its natural preorder by:
$$x\leq y :\Leftrightarrow \exists k\in A: x + k = y$$
Assume $\leq$ is antisymmetric.
What kind of "obvious" properties can one assume such that: $$ \bigvee_{i\in I} (x_i + y) = \bigvee_{i\in I} x_i + y$$ whenever these suprema exist?
Note that in any case we have: $$\bigvee_{i\in I} (x_i + y_i) \leq \bigvee_{i\in I} x_i + \bigvee_{i\in I} y_i$$
so the question comes down to when the other inequality holds.
Movitaion: The above property is necessary to show that “continuous monoids” are “complete”.