An extension of Wilf's conjecture to affine semigroups

TL;DR

This paper extends Wilf's conjecture from numerical semigroups to affine semigroups within rational cones, exploring their structure, generation, and properties, and verifying the conjecture for specific families of these semigroups.

Contribution

It introduces a framework for studying $ ext{C}$-semigroups, extends Wilf's conjecture to this broader context, and identifies families satisfying the extended conjecture.

Findings

Extended Wilf's conjecture to $ ext{C}$-semigroups.

Developed a method to generate the tree of $ ext{C}$-semigroups.

Identified families of $ ext{C}$-semigroups fulfilling the extended conjecture.

Abstract

Let $\CaC\subset \Q^p$ be a rational cone. An affine semigroup $S\subset \CaC$ is a $\CaC$-semigroup whenever $(\CaC\setminus S)\cap \N^p$ has only a finite number of elements. In this work, we study the tree of $\CaC$-semigroups, give a method to generate it and study their subsemigroups with minimal embedding dimension. We extend Wilf's conjecture for numerical semigroups to $\CaC$-semigroups and give some families of $\CaC$-semigroups fulfilling the extended conjecture. We also check that other conjectures on numerical semigroups seem to be also satisfied by $\CaC$-semigroups.