If $x + y \sim x$ in a commutative monoid, does this imply $y \sim 0$? Let $(M,0,+)$ be a commutative monoid. A congruence relation is an equivalence relation, such that
$$
 a \sim b, c \sim d \quad \mbox{implies} \quad a + c \sim b + d. 
$$
for all $a,b,c,d \in M$.

Fix some $x,y \in M$. Does $x + y \sim x$ imply $y \sim 0$? Do you know a counter-example?


Minimal counterexample: $(M, \oplus)$ with $M = \{0, 1, 2 \}$, $x \oplus y = \min \{ x+y , 2\}$ and $1 \sim 2$. Then $1 \sim (1 \oplus 1)$ but $1 \not\sim 0$.
