Numerical Semigroups with a Given Frobenius Number and Some Fixed Gaps

If P is a nonempty finite subset of positive integers, then A(P)={S∣S is a numerical semigroup, S∩P=∅ and max(P) is the Frobenius number of S}. In this work, we prove that A(P) is a covariety; therefore, we can arrange the elements of A(P) in the form of a tree. This fact allows us to present several algorithms, including one that calculates all the elements of A(P), another that obtains its maximal elements (with respect to the set inclusion order) and one more that computes the elements of A(P) that cannot be expressed as an intersection of two elements of A(P), that properly contain it.