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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.

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  • $\begingroup$ Conversely, there is also the notion of absorbing element for an element a such that for all b one has a = a + b. (When multiplicative notation is used this would 'also' be called zero element, for additive notation sometimes infinity-element). $\endgroup$
    – user9072
    Mar 11, 2014 at 16:16
  • $\begingroup$ I would just say b right stabilizes a or belongs to the right stabilizer of a. $\endgroup$
    – Benjamin Steinberg
    Mar 12, 2014 at 1:51

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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]

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