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.



Terminology for the equation $a=a+b$ in commutative semigroups
Tarski, cardinal algebras, pag. 10 writes:


Formulas of the type "$a+b=b$" [...] can be read "$a$ is absorbed by $b$" or "$b$ absorbs $a$". The relation of absorption plays an important role in the arithmetic of C.[ardinal]A.[lgebra]'s.


Wehrung, Injective positively ordered monoids I [ <http://www.sciencedirect.com/science/article/pii/002240499290104N> ] pag. 47 writes


$a <\!\!< b$ will always be the statement $a + b = b$.


[b.t.w. if you cite Wehrung be sure to not use the word "lattice" in your paper, or some referee might diagnose a severe lack of interaction with other parts of mathematics]


