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

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 $I\left(S\right)$-semigroup is the affine semigroup $P\cup \left\{0\right\}$, with $P$ an ideal of $S$. We characterise the $I\left(S\right)$-semigroups and the ones that also are ${C}$-semigroups. Moreover, some algorithms are provided to compute all the $I\left(S\right)$-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.