On ideals of affine semigroups and affine semigroups with maximal embedding dimension

TL;DR

This paper characterizes ideals of affine semigroups, introduces affine semigroups with maximal embedding dimension, and provides algorithms for their computation, advancing understanding of their structure and classification.

Contribution

It offers a complete characterization of $I(S)$-semigroups and affine semigroups with maximal embedding dimension, along with algorithms for their identification.

Findings

Characterization of $I(S)$-semigroups.

Introduction of affine semigroups with maximal embedding dimension.

Algorithms for computing specific classes of affine semigroups.

Abstract

Let $S\subseteq \mathbb N^p$ be a semigroup, any $P\subseteq S$ is an ideal of $S$ if $P+S\subseteq P$, and an $I(S)$-semigroup is the affine semigroup $P\cup \{0\}$, with $P$ an ideal of $S$. We characterise the $I(S)$-semigroups and the ones that also are $\mathcal C$-semigroups. Moreover, some algorithms are provided to compute all the $I(S)$-semigroups satisfying some properties. From a family of ideals of $S$, we introduce the affine semigroups with maximal embedding dimension, characterising them and describing some families.