  A numerical monoid is an additive submonoid of the non-negative integers.
Given a numerical monoid $S$, consider the family of "shifted" monoids $M_n$
obtained by adding $n$ to each generator of $S$. In this paper, we examine
minimal relations among the generators of $M_n$ when $n$ is sufficiently large,
culminating in a description that is periodic in the shift parameter $n$. We
explore several applications to computation, combinatorial commutative algebra,
and factorization theory.
