# On the computation of the Ap\'ery set of numerical monoids and affine semigroups

## TL;DR

This paper introduces a Groebner basis-based method for efficiently computing the Apéry set of numerical semigroups and affine semigroups, enabling analysis of their algebraic properties such as symmetry and Gorenstein conditions.

## Contribution

It presents a novel computational approach using Groebner bases for Apéry set calculation and extends it to affine semigroups, facilitating algebraic property verification.

## Key findings

- Efficient computation of Apéry sets using Groebner bases.
- Ability to determine the type set and Gorenstein condition.
- Generalization from numerical to affine semigroups.

## Abstract

A simple way of computing the Ap\'ery set of a numerical semigroup (or monoid) with respect to a generator, using Groebner bases, is presented, together with a generalization for affine semigroups. This computation allows us to calculate the type set and, henceforth, to check the Gorenstein condition which characterizes the symmetric numerical subgroups.