# Patterns on the numerical duplication by their admissibility degree

## TL;DR

This paper develops a theory of patterns on numerical semigroups based on admissibility degree, exploring their properties, equivalences, and extensions to numerical duplication and rings.

## Contribution

It introduces the concept of admissibility degree for patterns on numerical semigroups and characterizes patterns equivalent to the Arf pattern, extending the theory to numerical duplication and rings.

## Key findings

- Arf pattern induces all strongly admissible patterns
- Characterization of patterns equivalent to the Arf pattern
- Analysis of patterns on numerical duplication for large d

## Abstract

We develop the theory of patterns on numerical semigroups in terms of the admissibility degree. We prove that the Arf pattern induces every strongly admissible pattern, and determine all patterns equivalent to the Arf pattern. We study patterns on the numerical duplication $S \Join^d E$ when $d \gg0$. We also provide a definition of patterns on rings.