Properties and applications of the Ap\'ery set of good semigroups in $\mathbb{N}^d$

TL;DR

This paper explores the properties and applications of the Apéry set of good semigroups in multidimensional integer lattices, focusing on duality, non-local cases, and value semigroups of plane curves.

Contribution

It extends the understanding of Apéry sets for good semigroups, including duality properties and applications to non-local and plane curve semigroups.

Findings

Duality of symmetric and almost symmetric good semigroups analyzed.

Apéry sets of non-local good semigroups characterized.

Applications to value semigroups of plane curves demonstrated.

Abstract

In this article we discuss some applications of the construction of the Ap\'ery set of a good semigroup in $\mathbb{N}^d$ given in the previous paper [Partition of the complement of good semigroup ideals and Ap\'ery sets, Communications in Algebra, 49, No. 10, 4136-4158 (2021))]. In particular we study: the duality of a symmetric and almost symmetric good semigroup, the Ap\'ery set of non-local good semigroups and the Ap\'ery set of value semigroups of plane curves.