Uniqueness of two side zeroes of binary operation

Question

I came across the following fact in group theory:

Two-sided identity of binary operation is unique.

Does the similar statement for two sided zero also holds? :

Two-sided zero of binary operation is unique.

I feel yes. Because say if we have two distinct two sided zeroes $y$ and $z$ for operation $*$, then we will have,

$y*z=y$ ($y$ being zero)
$y*z=z$ ($z$ being zero)

Does this implies $y=z$. Is this right way to reason out with?

Answer 1

You are right. If $0, 0'$ are two two-sided zeros, then

$$\underbrace{0=00'}_{0\text{ a left zero}}=\overbrace{00'=0'}^{0'\text{ a right zero}},$$

meaning that such zeros are unique by transitivity.

You had this yourself: you just needed transitivity of equality.

To learn more about such things, consider Semigroup Theory or Magma Theory.