讲座内容：

Let $S$ be a numerical semigroup, that is, a cofinite submonoid of the non-negative integers under addition. A non-empty set of integers $I$ is said to be an ideal of $S$ if $I+S\subseteq I$ and $I$ has a minimum. The sum of two ideals $I$ and $J$, defined as $I+J=\{i+j : i\in I, j\in J\}$, is also an ideal of $S$. Thus, the set of ideals of $S$, denoted $\mathfrak{I}(S)$, is a commutative monoid under this operation, with neutral element $S$. If $S$ and $T$ are numerical semigroups and $\mathfrak{I}(S)$ is isomorphic to $\mathfrak{I}(T)$, then $S$ and $T$ must be the same numerical semigroup.

If $I$ and $J$ are ideals of $S$, we write $I\sim J$ if $I=z+J$ for some $z \in \mathbb Z$. The ideal class monoid of $S$ is defined as the set of ideals of $S$ modulo this relation, where addition of two classes $[I]$ and $[J]$ is defined as $[I]+[J] = [I+J]$.

An ideal $I$ is said to be normalized if $\min(I) = 0$. The set of normalized ideals of $S$, denoted by $\mathfrak{I}_0(S)$, is a monoid isomorphic to the ideal class monoid of $S$ \cite{icm}.  It is known that if $S$ and $T$ are numerical semigroups for which $\mathfrak{I}_0(S)$ is isomorphic to $\mathfrak{I}_0(T)$, then $S$ and $T$ must be equal.

The set $\mathfrak{I}_0(S)$ becomes a poset under inclusion. In \cite{iso-icm}, we also prove that if $S$ and $T$ are numerical semigroups such that the poset $(\mathfrak{I}_0(S),\subseteq)$ is isomorphic to the poset $(\mathfrak{I}_0(T),\subseteq)$, then $S = T$.

On $\mathfrak{I}_0(S)$ we can define a partial order $\preceq$ as $I\preceq J$ if there exists $K \in \mathfrak{I}_0(S)$ such that $I+K = J$. We know that if $S$ and $T$ are numerical semigroups with multiplicity three such that the poset $(\mathfrak{I}_0(S),\preceq)$ is isomorphic to the poset $(\mathfrak{I}_0(T),\preceq)$, then $S = T$. However, if we remove the condition on the multiplicity, this isomorphism problem is still open.

In recent work with Bonzio, we study the case when the poset $(\mathfrak{I}_0(S),\preceq)$ is a lattice. We show that this is the case if and only if the multiplicity of $S$ is at most four.

During the talk, will give an overview of these recent results and present some open problems.