主 讲 人 ：Pedro A. García-Sánchez    教授

活动时间：2025-09-21 16:20:00

地      点 ：数学科学学院D203报告厅(Zoom: https://us06web.zoom.us/j/86763384947?pwd=qXhOzOcHvaaiw2jqADI8iqNavdgm14.1, ID: 86763384947; passcode: 612593)

主办单位：数学科学学院

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.

主讲人介绍：

Pedro A. García-Sánchez defended his PhD thesis at the University of Granada (Spain) in 1996, where he has held a permanent position since 1999, became a full professor of algebra in 2017, and has served as the Director for Internationalization at the International School for Postgraduate Studies and coordinator of the Master in Mathematics. His primary research interests include numerical semigroups, commutative monoids, and non-unique factorization. He has served as the principal investigator for several research projects in these areas, and has authored or coauthored four books and approximately one hundred research papers. He is an associate editor of Communications in Algebra and has actively participated in various teaching innovation projects. He is currently vice-dean of Quality and Teaching Innovation at the Faculty of Sciences of the University of Granada.