On some numerical semigroup transforms

TL;DR

This paper introduces a new numerical semigroup transform that preserves key invariants related to Wilf's conjecture and organizes certain semigroups into rooted trees, comparing it with a similar existing transform.

Contribution

It presents a novel semigroup transform that maintains invariants relevant to Wilf's conjecture and analyzes its effects on the structure and properties of numerical semigroups.

Findings

Transform fixes invariants except embedding dimension

Semigroups form rooted trees under the transform

Insights into Wilf's conjecture implications

Abstract

In this paper we introduce a particular semigroup transform $\mathcal{A}$ that fixes the invariants involved in Wilf's conjecture, except the embedding dimension. It also allows one to arrange the set of not ordinary and not irreducible numerical semigroups in a family of rooted trees. We study also another transform, having similar features, that has been introduced by Bras-Amor\'os, and we make a comparison of them. In particular we study the behaviour of the embedding dimension under the action of such transforms, providing some consequences concerning Wilf's conjecture.