Greedy Sets and Greedy Numerical Semigroups

TL;DR

This paper extends the concept of greediness from change-making problems to sets and numerical semigroups, providing algorithms and conditions to identify greedy structures, especially for small sets and specific generated semigroups.

Contribution

It introduces a new notion of greediness for sets and numerical semigroups, along with algorithms and characterizations for particular cases.

Findings

An algorithm to determine if a set is greedy.

Conditions for sets of size three to be greedy.

Numerical semigroups generated by three consecutive integers are greedy.

Abstract

Motivated by the change-making problem, we extend the notion of greediness to sets of positive integers not containing the element $1$, and from there to numerical semigroups. We provide an algorithm to determine if a given set (not necessarily containing the number $1$) is greedy. We also give specific conditions for sets of cardinality three, and we prove that numerical semigroups generated by three consecutive integers are greedy.