An extension of the Wilf conjecture to semimodules over a numerical semigroup

TL;DR

This paper extends the Wilf conjecture to semimodules over numerical semigroups by introducing a new invariant called the Wilf function, providing a novel approach to understanding and potentially proving the conjecture.

Contribution

It introduces the Wilf function for semigroups and semimodules, establishing new relations and proposing an alternative conjecture that implies the original Wilf conjecture.

Findings

Wilf function defined for semigroups and semimodules

Relation between Wilf functions of semigroups and associated semimodules

Proposed alternative conjecture implying the Wilf conjecture

Abstract

The aim of this paper is to propose an extension of the Wilf conjecture to semimodules over a numerical semigroup through a new approach toward the solution of the Wilf conjecture on numerical semigroups. The key point is the introduction of a new invariant that we call Wilf function of a semigroup that can be also defined for a semimodule over the semigroup. We also study the relation of the Wilf function of a semigroup and the Wilf function of certain semimodules associated to a gap of the semigroup. As a consequence, we obtain an alternative conjecture which also implies the Wilf conjecture. Finally, we show that the natural generalization of Wilf conjecture to semimodules over the semigroup does not work unless we used the Wilf function of the semimodule. In this direction, we propose several questions regarding the possible extensions of Wilf-type inequalities for semimodules.