Ordinarization Transform of a Numerical Semigroup and Semigroups with a Large Number of Intervals

TL;DR

This paper introduces the ordinarization transform for numerical semigroups, studies the resulting trees' structure, and explores the relationship between semigroup depth and the number of gap intervals, providing new insights into their enumeration.

Contribution

It defines the ordinarization transform, constructs semigroup trees by genus, analyzes their regularity, and links semigroup depth to the number of gap intervals with explicit characterizations.

Findings

Number of semigroups increases with genus at certain depths.

Semigroups at a given depth correspond to those with many gap intervals.

Conjecture that semigroup count increases with genus at all depths.

Abstract

A numerical semigroup is said to be ordinary if it has all its gaps in a row. Indeed, it contains zero and all integers from a given positive one. One can define a simple operation on a non-ordinary semigroup, which we call here the ordinarization transform, by removing its smallest non-zero non-gap (the multiplicity) and adding its largest gap (the Frobenius number). This gives another numerical semigroup and by repeating this transform several times we end up with an ordinary semigroup. The genus, that is, the number of gaps, is kept constant in all the transforms. This procedure allows the construction of a tree for each given genus containing all semigrpoups of that genus and rooted in the unique ordinary semigroup of that genus. We study here the regularity of these trees and the number of semigroups at each depth. For some depths it is proved that the number of semigroups increases with the genus and it is conjectured that this happens at all given depths. This may give some lights to a former conjecture saying that the number of semigroups of a given genus increases with the genus. We finally give an identification between semigroups at a given depth in the ordinarization tree and semigroups with a given (large) number of gap intervals and we give an explicit characterization of those semigroups.