Quasi-ordinarization transform of a numerical semigroup

TL;DR

This paper introduces the quasi-ordinarization transform for numerical semigroups, organizing them into a forest structure to explore conjectures about their enumeration and properties.

Contribution

It presents the quasi-ordinarization transform, linking it to existing transforms, and offers a new framework for studying numerical semigroups of fixed genus.

Findings

Organizes semigroups into a forest structure based on genus.

Provides properties and relations of the quasi-ordinarization transform.

Proposes new conjectures on the enumeration of semigroups.

Abstract

We introduce the quasi-ordinarization transform of a numerical semigroup. This transform will allow to organize all the semigroups of a given genus in a forest rooted at all quasi-ordinary semigroups with the given genus. This construction provides an alternative approach to the conjecture on the increasingness of the number of numerical semigroups for each given genus. We elaborate on the number of nodes at each tree depth in the forest and present a few new conjectures that can be developed in the future. We prove some properties of the quasi-ordinarization transform, its relations with the ordinarization transform, and we also present an alternative approach to the conjecture that the number of numerical semigroups of each given genus is increasing.