Absorbing element

Remarks

Example: in $(\Z, \cdot)$ the integer $0$ is absorbing, because $0 \cdot a = a \cdot 0 = 0$ for all $a \in \Z$.

Non-example: in $(\Z, +)$ the integer $0$ is not absorbing, because $0 + 1 = 1 \ne 0$; it is an identity element instead.

If $z$ is left absorbing and $z'$ is right absorbing, then $z = z \ast z' = z'$. So an operation has at most one absorbing element. On a set with at least two elements, the operation $a \ast b = a$ makes every element left absorbing and no element right absorbing.