Speaker: Caleb Shor, Western New England University

Title: Characterization of free numerical semigroups

Abstract: Let $\mathbb{N}_0$ denote the set of nonnegative integers. A numerical semigroup $S$ is a subset of $\mathbb{N}_0$ that is closed under addition, contains 0, and has finite complement in $\mathbb{N}_0$. Such objects arise naturally in algebraic geometry and number theory. They’re also of interest to fans of postage stamps and chicken nuggets.

Given a set $G\subset\mathbb{N}_0$ with $\gcd(G)=1$, the set of nonnegative linear combinations of elements of $G$ is a numerical semigroup. It happens that every numerical semigroup arises this way, so we can think of numerical semigroups in terms of their generating sets. For example, if $G=\{3,5\}$, then $S=\{3x+5y : x,y\in\mathbb{N}_0\}=\mathbb{N}_0\setminus\{1,2,4,7\}$. In general, we will have $S=\mathbb{N}_0\setminus \textit{NR}(G)$, where $\textit{NR}(G)$ is the set of integers not representable in terms of $G$.

Numerous questions arise naturally. For instance, given a set $G$, what can we say about the cardinality, largest element, structure, and other associated objects and properties of $\textit{NR}(G)$? For a general set $G$, these can be difficult questions. In this talk, we will consider “free” numerical semigroups, which are generated by sets $G$ with certain smoothness conditions. In particular, we will see an identity which completely characterizes $\textit{NR}(G)$, and we will use that to answer some of the questions mentioned above.