Let $(S,+)$ be a commutative semigroup. For $a,b\in S$ consider the equation $a=a+b$. Does such a relation between the given $a$ and $b$ have a name? I am currently using such equations quite often and I would like to find the appropriate references and basic facts. I have an impression seeing a similar subject in MathOverflow recently, but it is quite hard to find something when you don't know how it is called.

Surely, if $(\forall a\in S)\ a=a+b$ then we say $b$ is a *neutral element in $S$* (a *zero* in the case of the additive notation). But what about the other cases?

In the case that there is no name for the relation $a=a+b$, I would suggest to say either that *$b$ is a relative zero (with respect) to $a$* or that *$b$ is a local zero (with respect) to $a$*. Could somebody give me a hint or at least an opinion? Thanks.

multiplicativenotation is used this would 'also' be called zero element, for additive notation sometimes infinity-element). $\endgroup$ – user9072 Mar 11 '14 at 16:16