Is there a definition/name for such monoids?
Has any theory been developed for such monoids?
Are there any any references/links delving into this with perhaps additional axioms?
2026-03-25 22:04:49.1774476289
Commutative monoids $(S,+,0)$ where $\forall x,y \in S$ $x+ y = x$ implies $y = 0$?
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One algebraic structure which might fit the bill is that of a cancellative monoid. This is a special case of a cancellative semigroup: https://en.wikipedia.org/wiki/Cancellative_semigroup. In particular, if $(S,+,0)$ is our monoid, we have the cancellative law: for all $x,y,z\in S$, $$ x+y=x+z \implies y=z$$ The property $x+y=x$ implies $y=0$ is a consequence of this. I am personally unaware of a monoid with just the property $x+y=x$ implies $y=0$.