Patterns of ideals of numerical semigroups

TL;DR

This paper introduces a unified framework for patterns of ideals in numerical semigroups, providing new insights into their structure and the patterns they admit, with broad implications for the theory of numerical semigroups.

Contribution

It unifies previous definitions of patterns of ideals in numerical semigroups and explores their structural properties and the sets of patterns they admit.

Findings

Characterization of the image of patterns of ideals

Structural results on sets of patterns admitted by ideals

General interest results on the structure of numerical semigroup ideals

Abstract

This article introduces patterns of ideals of numerical semigroups, thereby unifying previous definitions of patterns of numerical semigroups. Several results of general interest are proved. More precisely, this article presents results on the structure of the image of patterns of ideals, and also on the structure of the sets of patterns admitted by a given ideal.