Good subsemigroups of $\mathbb N^n$

TL;DR

This paper studies good subsemigroups of a5^n, providing a combinatorial framework, defining minimal generators, and offering methods to compute canonical ideals and Arf closures for the case n=2.

Contribution

It introduces the concept of good systems of generators and proves their minimality is unique, advancing the combinatorial understanding of good semigroups.

Findings

Minimal good systems of generators are unique.

Provides a constructive method to compute the canonical ideal.

Offers a way to compute the Arf closure for n=2.

Abstract

Value semigroups of non irreducible singular algebraic curves and their fractional ideals are submonoids of $\mathbb Z^n$ that are closed under infimums, have a conductor and fulfill a special compatibility property on their elements. Monoids of $\mathbb N^n$ fulfilling these three conditions are known in the literature as good semigroups and there are examples of good semigroups that are not realizable as the value semigroup of an algebraic curve. In this paper we consider good semigroups independently from their algebraic counterpart, in a purely combinatoric setting. We define the concept of good system of generators, and we show that minimal good systems of generators are unique. Moreover, we give a constructive way to compute the canonical ideal and the Arf closure of a good subsemigroup when $n=2$.