Let $M$ be a monoid and write its multiplication as $+$. Some monoids have the property that for each $k\in M$ the set $\{(i,j) \in M\times M \mid i + j = k\}$ is finite. Is there a name for this proeprty?
An example is the natural numbers with addition: the set $\{(i,j) \in \mathbb{N}\times \mathbb{N} \mid i + j = k\}$ is finite for each $k\in \mathbb{N}$. A non-example is the reals with addition, or the integers with addition.
As a bonus question, is there a name for monoids where this set is at most countable?