  Numerical monoids (cofinite, additive submonoids of the non-negative
integers) arise frequently in additive combinatorics, and have recently been
studied in the context of factorization theory. Arithmetical numerical monoids,
which are minimally generated by arithmetic sequences, are particularly
well-behaved, admitting closed forms for many invariants that are difficult to
compute in the general case. In~this paper, we answer the question "when does
omitting generators from an arithmetical numerical monoid $S$ preserve its
(well-understood) set of length sets and/or Frobenius number?" in two extremal
cases: (i) we characterize which individual generators can be omitted from $S$
without changing the set of length sets or Frobenius number; and (ii) we prove
that under certain conditions, nearly every generator of $S$ can be omitted
without changing its set of length sets or Frobenius number.
