In this paper we study those submonoids of \(\mathbb {N}^d\) with a nontrivial pseudo-Frobenius set. In the affine case, we prove that they are the affine semigroups whose associated algebra over a field has maximal projective dimension possible. We prove that these semigroups are a natural generalization of numerical semigroups and, consequently, most of their invariants can be generalized. In the last section we introduce a new family of submonoids of \(\mathbb {N}^d\) and using its pseudo-Frobenius elements we prove that the elements in the family are direct limits of affine semigroups.

中文翻译：

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关于$$ \ mathbb {N} ^ d $$ Nd的子类生物的伪Frobenius元素

在本文中，我们研究了具有非平凡伪Frobenius集的\（\ mathbb {N} ^ d \）的那些子monoid 。在仿射情况下，我们证明它们是仿射半群，其在场上的相关代数具有最大可能的投影维数。我们证明这些半群是数值半群的自然概括，因此，它们的大多数不变量都可以被概括。在上一节中，我们介绍了\（\ mathbb {N} ^ d \）的一个新的亚类族，并使用其伪Frobenius元素证明了该族中的元素是仿射半群的直接限制。