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arXiv · 1909.13120

A generalization of Wilf's conjecture for Generalized Numerical Semigroups

Carmelo Cisto·Michael DiPasquale·Gioia Failla·Zachary Flores·Chris Peterson·Rosanna Utano
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Abstract

A numerical semigroup is a submonoid of $\mathbb N$ with finite complement in $\mathbb N$. A generalized numerical semigroup is a submonoid of $\mathbb{N}^{d}$ with finite complement in $\mathbb{N}^{d}$. 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 $\mathcal C$-semigroups, was proposed by Garc\'ia-Garc\'ia, Mar\'in-Arag\'on, 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 Garc\'ia-Garc\'ia, Mar\'in-Arag\'on, and Vigneron-Tenorio.

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Carmelo Cisto, Michael DiPasquale, Gioia Failla, Zachary Flores, Chris Peterson, Rosanna Utano. 2019-09-28. A generalization of Wilf's conjecture for Generalized Numerical Semigroups. https://arxiv.org/abs/1909.13120

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