Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Some Families of Greedy Numerical Semigroups

Arnau Messegué-Buisan, Hebert Pérez-Rosés

Abstract

The change-making problem was recently extended to sets of positive integers not containing the element $1$, and from there to numerical semigroups. A greedy numerical semigroup is defined as a numerical semigroup where the greedy representation of an element is optimal with respect to the number of summands. In this paper we identify some new families of greedy numerical semigroups.

Some Families of Greedy Numerical Semigroups

Abstract

The change-making problem was recently extended to sets of positive integers not containing the element , and from there to numerical semigroups. A greedy numerical semigroup is defined as a numerical semigroup where the greedy representation of an element is optimal with respect to the number of summands. In this paper we identify some new families of greedy numerical semigroups.
Paper Structure (4 sections, 17 theorems, 38 equations, 2 algorithms)

This paper contains 4 sections, 17 theorems, 38 equations, 2 algorithms.

Table of Contents

  1. Greedy sets and greedy numerical semigroups
  2. Greedy semigroups of embedding dimension three
  3. Semigroup families generated by the Fibonacci sequence
  4. Some open problems

Key Result

theorem 1

As in Definition def:counterexample let $S=\{ s_1, \ldots, s_t \}$, with $t \geq 3$, be a set of generators such that $1 \leq s_1 < s_2 < \ldots < s_t$ and $\gcd(s_1, \ldots, s_t) = 1$. Then, $S$ is greedy if, and only if, $S$ does not have any counterexample $k$ in the interval where $F(\mathbb{S})$ denotes the Frobenius number of $\mathbb{S}$.

Theorems & Definitions (24)

  • definition 1
  • definition 2
  • definition 3
  • theorem 1: PR25
  • definition 4
  • theorem 2: PR25
  • lemma 1: PR25
  • theorem 3: PR25
  • lemma 2
  • proposition 1
  • ...and 14 more