arfClosure -- Compute the Arf closure of a numerical semigroup

A numerical semigroup S is \emph{Arf} if for every triple of elements x \geq y \geq z in S, the element x+y-z also lies in S. Equivalently, S is Arf if and only if every semigroup in the sequence of blowups (the infinitely near semigroups of S) has minimal multiplicity, that is, has multiplicity equal to its embedding dimension.

The \emph{Arf closure} Arf(S) is the smallest Arf semigroup containing S. The function arfClosure returns a list of minimal generators of Arf(S).

The Arf closure is computed from the multiplicities of the infinitely near semigroups of S: if those multiplicities are m_0, m_1, ..., m_(r-1), then the partial sums m_0, m_0+m_1, ..., m_0+...+m_(r-1) lie in Arf(S), and together with the integers immediately above the largest partial sum (up to m_0 - 1 of them) they generate Arf(S).

The third example is already Arf, so the Arf closure equals the input semigroup (up to choice of minimal generators). The trivial semigroup N (generated by 1) is its own Arf closure: