Decompositions of ideals into irreducible ideals in numerical semigroups

It is proved that each ideal I of a numerical semigroup S is in a unique way a finite irredundant intersection of irreducible ideals. The same result holds if "irreducible ideals" are replaced by "Z-irreducible ideals". The two decompositions are essentially different and, if n(I ) and N (I ) respectively are the number of irreducible or Z-irreducible components, it is n(I ) ≤ N (I ) ≤ e, where e is the multiplicity of S . However, if I is a principal ideal, then n(I ) = N (I ) = t, where t is the type of S .