© 2024 the author(s), published by De Gruyter.Let S N p Ssubseteq {{mathbb{N}}}^{p} be a semigroup, any P S Psubseteq S is an ideal of S S if P + S P P+Ssubseteq P, and an I (S) Ileft(S) -semigroup is the affine semigroup P { 0 } Pcup left{0 ight}, with P P an ideal of S S. We characterise the I (S) Ileft(S) -semigroups and the ones that also are C {mathcal{C}} -semigroups. Moreover, some algorithms are provided to compute all the I (S) Ileft(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.