On some classes of generalized numerical semigroups

TL;DR

The paper extends Wilf-type questions to generalized numerical semigroups (GNS) in $\mathbb{N}^d$ by introducing three natural classes: $T$-stripe GNSs, $T$-graded GNSs, and Axis GNSs. It proves a general sufficient condition for the generalized Wilf's conjecture, $e(S)\ge d(t(S)+1)$, and shows this condition holds for all $T$-stripe GNSs, providing explicit invariant formulas and exploring stronger bounds in special cases. It also derives structural results and partial Wilf-type conclusions for $T$-graded and Axis GNSs, including explicit formulas in the two-generator base $T=\langle m,n\rangle$ and a direct relationship between base semigroup properties and the high-dimensional constructions. The work connects base numerical semigroup invariants to high-dimensional invariants, offering constructive frameworks and open problems for further generalizations and computational exploration. Overall, it advances the understanding of when Wilf-type inequalities extend to multidimensional semigroups and highlights several rich directions for future research.

Abstract

A generalized numerical semigroup is a submonoid of $\mathbb{N}^d$ with finite complement in it. In this work we study some properties of three different classes of generalized numerical semigroups. In particular, we prove that the first class satisfies a generalization of Wilf's conjecture, by introducing a generalization of a well-known sufficient condition for Wilf's conjecture in numerical semigroups, that involves the type of the semigroup. Partial results for Wilf's generalized conjecture are obtained also for the other two classes, and some open questions are provided.

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Theorems & Definitions (52)

• Conjecture 2.1: Generalized Wilf's conjecture
• Proposition 2.2
• proof
• Corollary 2.3
• Lemma 2.4
• proof
• Example 2.5
• Definition 3.1
• Proposition 3.2
• proof