The present paper studies the pseudo-Frobenius numbers from submonoids of \(\mathbb{N}^d\) making emphasis in \(\mathcal{C}\)-semigroups, i.e., affine semigroups where the cardinality of the complement of the semigroup in the cone spanned by itself is finite.
The authors introduce the maximal projective dimension semigroups (MPD-semigroups) and prove that a semigroup \(S\) is a MPD-semigroup if and only if it has almost one pseudo-Frobenius number and, in this case, the cardinality of the set of all pseudo-Frobenius numbers of \(S\) is finite. They also study the gluing of this kind of semigroups and show that it is also a MPD-semigroup under some assumptions about their pseudo-Froebnius number. Another interesting result is that a \(\mathcal{C}\)-semigroup is irreducible if the set of pseudo-Frobenius number has a specific form.
They finish the paper with a new family of semigroups: the principal ideal monoids (PI-monoids) which are of the form \(a+T\) with \(a\in T\) and \(T\) a monoid. Some general propositions about them and a characterization are shown. This characterization allows to make a relation between PI-monoids and MPD-semigroups.