AI 中文总结
本文研究数值半群的树结构，提出构造固定重数、弗罗贝尼乌斯数或亏格的内部半群的算法，猜想内部半群数量多于叶半群，还给出同时固定重数与弗罗贝尼乌斯数时两类半群数量的部分闭式公式。
AI 中文摘要
本文研究数值半群的树结构，内部数值半群是位于该树内部节点的半群，类似地，叶数值半群位于叶节点。研究具有固定重数、弗罗贝尼乌斯数或亏格的内部半群，提供构造所有此类半群的算法。建立若干猜想，例如在固定重数、固定弗罗贝尼乌斯数、固定亏格这三种情形下，结果均表明内部数值半群的数量总是多于叶数值半群。最后研究同时具有固定重数和弗罗贝尼乌斯数的数值半群，针对重数和弗罗贝尼乌斯数的某些取值，还给出了内部半群与叶半群数量的闭式计数公式。
英文摘要
In this paper the tree structure of numerical semigroups is studied. An internal numerical semigroup is a semigroup located in an internal node of the tree. Analogous a leaf numerical semigroup is placed in a leaf node. Internal semigroups with fixed multiplicity, Frobenius number or genus are studied by providing algorithms to construct all of them. Several conjectures are established, for example, in each case (fixed multiplicity, fixed Frobenius number and fixed genus respectively), the results suggest that there are always more internal than leaf numerical semigroups. Finally, numerical semigroups with fixed multiplicity and Frobenius number simultaneously are investigated. In this case with two invariants fixed, moreover closed formulas to count the number of internal and leaf semigroups are provided for some values of multiplicity and Frobenius number.
Comments23 pages for main paper + 16 pages of appendix with data
