Gapset Extensions, Theory and Computations

TL;DR

This paper extends set theoretic concepts to sub-semigroups of natural numbers, characterizes gapsets for efficient computation of numerical semigroups, and proves a non-decreasing sequence property related to gapsets.

Contribution

It introduces the extension of gapsets and provides a proof that the number of gapsets of size g is non-decreasing, advancing understanding of their structure.

Findings

Characterized gapsets for numerical semigroups

Developed a more efficient computational approach

Proved the non-decreasing sequence of gapsets of size g

Abstract

In this paper we extend some set theoretic concepts of numerical semigroups for arbitrary sub-semigroups of natural numbers. Then we characterized gapsets which leads to a more efficient computational approach towards numerical semigroups and finally we introduce the extension of gapsets and prove that the sequence of the number of gapsets of size $g$ is non-decreasing as a weak version of Bras-Amor\'os's conjecture.