# Elasticity in Apery sets

## TL;DR

This paper investigates the properties of Apery sets in numerical semigroups, focusing on the non-uniqueness of element factorizations and introducing new invariants to measure this phenomenon.

## Contribution

It introduces new invariants to quantify non-unique factorizations within Apery sets of numerical semigroups, advancing understanding of their algebraic structure.

## Key findings

- Identification of elements with non-unique factorizations in Apery sets
- Development of new invariants to measure factorization non-uniqueness
- Insights into the structure of numerical semigroups through these invariants

## Abstract

A numerical semigroup $S$ is an additive subsemigroup of the non-negative integers, containing zero, with finite complement. Its multiplicity $m$ is its smallest nonzero element. The Apery set of $S$ is the set $\text{Ap}(S) = \{n \in S : n-m \notin S\}$. Fixing a numerical semigroup, we ask how many elements of its Apery set have nonunique factorization, and define several new invariants.