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

Abstract

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