Good subsemigroups of \({\mathbb {N}}^d\) have been introduced as the most natural generalization of numerical ones. Although their definition arises by taking into account the properties of value semigroups of analytically unramified rings (for instance the local rings of an algebraic curve), not all good semigroups can be obtained as value semigroups, implying that they can be studied as pure combinatorial objects. In this work, we are going to introduce the definition of length and genus for good semigroups in \({\mathbb {N}}^d\). For \(d=2\), we show how to count all the local good semigroups with a fixed genus through the introduction of the tree of local good subsemigroups of \({\mathbb {N}}^2\), generalizing the analogous concept introduced in the numerical case. Furthermore, we study the relationships between these elements and others previously defined in the case of good semigroups with two branches, as the type and the embedding dimension. Finally, we show that an analogue of Wilf’s conjecture fails for good semigroups in \({\mathbb {N}}^2\).

中文翻译：

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$$ {\ mathbb {N}} ^ 2 $$ N 2中的良半群树和Wilf猜想的推广

引入了\（{\ mathbb {N}} ^ d \）的好子半群，作为数值数的最自然的概括。尽管它们的定义是通过考虑分析无分支环的值半群（例如代数曲线的局部环）的性质而产生的，但并非所有好的半群都可以作为值半群获得，这意味着它们可以作为纯组合对象来研究。 。在这项工作中，我们将介绍\（{\ mathbb {N}} ^ d \）中好的半群的长度和类的定义。对于\（d = 2 \），我们介绍如何通过引入\（{\ mathbb {N}} ^ 2 \）的局部优良子半群的树来计算具有固定属的所有局部优良半群。，概括了在数字案例中引入的类似概念。此外，我们研究了在具有两个分支的良好半群的情况下，这些元素与先前定义的其他元素之间的关系，即类型和嵌入维数。最后，我们证明了对于（\（{\ mathbb {N}} ^ 2 \）中的好的半群，Wilf猜想的一个类似物失败了。