A generalization of Wilf's conjecture for Generalized Numerical Semigroups

TL;DR

This paper extends Wilf's conjecture to generalized numerical semigroups in higher dimensions, proving it for several large classes and exploring its relation to previous generalizations.

Contribution

It introduces a new generalization of Wilf's conjecture for generalized numerical semigroups and proves it for key classes like irreducible, symmetric, and monomial cases.

Findings

Proved the conjecture for irreducible semigroups

Validated the conjecture for symmetric semigroups

Confirmed the conjecture for monomial semigroups

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.