  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.
