A numerical semigroup is a submonoid of $${\mathbb {N}}$$ N with finite complement in $${\mathbb {N}}$$ N . A generalized numerical semigroup is a submonoid of $${\mathbb {N}}^{d}$$ N d with finite complement in $${\mathbb {N}}^{d}$$ N d . In the context of numerical semigroups, Wilf’s conjecture is a long standing open problem whose study has led to new mathematics and new ways of thinking about monoids. A natural extension of Wilf’s conjecture, to the class of $${\mathcal {C}}$$ C -semigroups, was proposed by García-García, Marín-Aragón, and Vigneron-Tenorio. In this paper, we propose a different generalization of Wilf’s conjecture, to the setting of generalized numerical semigroups, and prove the conjecture for several large families including the irreducible, symmetric, and monomial case. We also discuss the relationship of our conjecture to the extension proposed by García-García, Marín-Aragón, and Vigneron-Tenorio.

中文翻译：

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广义数值半群的威尔夫猜想的推广

数值半群是 $${\mathbb {N}}$$ N 的子幺半群，在 $${\mathbb {N}}$$ N 中有有限补。广义数值半群是 $${\mathbb {N}}^{d}$$ N d 的子幺半群，在 $${\mathbb {N}}^{d}$$ N d 中有有限补。在数值半群的背景下，威尔夫猜想是一个长期存在的悬而未决的问题，它的研究导致了新的数学和关于幺半群的新思维方式。García-García、Marín-Aragón 和 Vigneron-Tenorio 提出了 Wilf 猜想的自然扩展，即 $${\mathcal {C}}$$ C -半群的类。在本文中，我们提出了 Wilf 猜想的不同推广，用于广义数值半群的设置，并证明了几个大族的猜想，包括不可约、对称和单项式情况。