(Warning: This might be a silly question.) Suppose we have a commutative and associative binary operator $\cdot$ over a set of elements $S$. If $a \cdot t = b\cdot t$ for all $t\in S$, does $a=b$ necessarily? Intuitively I would say "yes", because in the world of this binary operator/set, $a$ and $b$ have the exact same properties. But I'm not sure what being "equal" entails in this context.
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