Example

(1) Every natural number is cancellable under addition; every natural number $\neq 0$ is cancellable under multiplication.

(2) In a lattice there can be no cancellable element under the law sup other than the identity element (least element) if it exists; similarly for inf. In particular, in the set of subsets of a set $E$, $\varnothing$ is the only cancellable element under the law $\cup$ and $E$ the only cancellable element under the law $\cap$.