On arithmetical numerical monoids with some generators omitted

TL;DR

This paper investigates conditions under which omitting generators from arithmetical numerical monoids preserves key invariants like the set of length sets and Frobenius number, providing characterizations and extremal cases.

Contribution

It characterizes when generators can be omitted from arithmetical numerical monoids without altering important invariants, advancing understanding of their structural properties.

Findings

Identifies generators whose omission does not change invariants.

Proves that nearly all generators can be omitted under certain conditions.

Provides a complete characterization for extremal cases.

Abstract

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.