The maximum lateness is at most $D$ exactly if some schedule meets the deadlines $\bar d_j = d_j + D$; a schedule that meets them has makespan at most the largest of them. So the decision version of a problem minimizing the maximum lateness without deadlines reduces to that of the problem with the deadlines $d_j + D$ that minimizes the makespan, with the largest deadline as threshold. If some $d_j + D$ is negative, or there are no jobs and $D$ is negative, the threshold is $-1$.

A specification $s$ of the family with operations, or of the batching family, reduces to $t$ through deadlines if $s$ minimizes the maximum lateness without deadlines and the specification obtained by giving it deadlines and the makespan as objective is field by field a special case of $t$. The relation is decidable.

Theorem. In both families, a specification that reduces to another through deadlines describes a problem whose decision version reduces to that of the other.