# Factoring in Embedding Dimension Three Numerical Semigroups

### Abstract

Let us consider a $3$-*numerical semigroup* $S=\langle{a,b,N}\rangle$. Given $m\in S$, the triple $(x,y,z)\in\mathbb{N}^3$ is a *factorization* of $m$ in $S$ if $xa+yb+zN=m$. This work is focused on finding the full set of factorizations of any $m\in S$ and as an application we compute the *catenary degree* of $S$. To this end, we relate a 2D tessellation to $S$ and we use it as a main tool.