Definition:Perfect Digit-to-Digit Invariant/Also defined as

Definition
Some sources disallow the zero from use in the definition of a perfect digit-to-digit invariant.

This is because $0^0$ is more usually defined defined as $0^0 = 1$ rather than as $0^0 = 0$.

If we define $0^0 = 1$ then not only does $438 \, 579 \, 088$ no longer qualify as a perfect digit-to-digit invariant, then nor does $0$ itself.