Apéry集与数值半群的理想类幺半群
Apéry sets and the ideal class monoid of a numerical semigroup
摘要：
本文研究了由V.Barucci和F. Khouja。证明了$\mathcal{C}\ell(S)$基数的新的界。我们观察到$\mathcal{C}\ell(S)$同构于$S$的最小元素为0的理想幺半群，这有助于将$\mathcal{C}\ell(S)$与$S$的Apéry集和Kunz坐标联系起来。研究了$\mathcal{C}\ell(S)$的一些组合和代数性质，包括理想的约化数，以及$\mathcal{C}\ell(S)$关于包含和加法的Hasse图.从这些图中，我们可以恢复一些值得注意的不变量的半群。最后，我们证明了关于$(\mathcal{C}\ell(S),+)$的不可约元、原子、夸克和素数的一些结果。幂等理想与上半群一致，幂等夸克对应于半群的酉扩张。我们证明了一个数值半群是不可约的当且仅当$\mathcal{C}\ell(S)$至多有两个夸克。
英文摘要：
The aim of this article is to study the ideal class monoid $\mathcal{C}\ell(S)$ of a numerical semigroup $S$ introduced by V. Barucci and F. Khouja. We prove new bounds on the cardinality of $\mathcal{C}\ell(S)$. We observe that $\mathcal{C}\ell(S)$ is isomorphic to the monoid of ideals of $S$ whose smallest element is 0, which helps to relate $\mathcal{C}\ell(S)$ to the Apéry sets and the Kunz coordinates of $S$. We study some combinatorial and algebraic properties of $\mathcal{C}\ell(S)$, including the reduction number of ideals, and the Hasse diagrams of $\mathcal{C}\ell(S)$ with respect to inclusion and addition. From these diagrams we can recover some notable invariants of the semigroup. Lastly, we prove some results about irreducible elements, atoms, quarks and primes of $(\mathcal{C}\ell(S),+)$. Idempotent ideals coincide with over-semigroups and idempotent quarks correspond to unitary extensions of the semigroup. We show that a numerical semigroup is irreducible if and only if $\mathcal{C}\ell(S)$ has at most two quarks.
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