infinitelyNearSemigroups -- The sequence of blowup semigroups of a numerical semigroup

Let R be the semigroup ring of a numerical semigroup S with multiplicity m, and let \mathfrak{m} = (t^s : s \in S, s > 0) be its maximal ideal. The \emph{blowup} of R at \mathfrak{m} is the subring R[\mathfrak{m}/t^m] = R[t^{s-m} : s \in S, s > 0] of the fraction field frac R; this blowup is again a semigroup ring, namely the semigroup ring of the \emph{blowup semigroup} S', generated by m together with s-m for every nonzero s in S.

Iterating this construction produces an ascending sequence of semigroup rings R = R_0 \subseteq R_1 \subseteq R_2 \subseteq \dots, each obtained from the previous by blowing up at its maximal ideal, and a corresponding ascending sequence of semigroups S = S_0 \subseteq S_1 \subseteq S_2 \subseteq \dots. The sequence stabilizes after finitely many steps when S_r becomes the trivial semigroup N (generated by 1), at which point R_r is the integral closure of R in frac R.

The function infinitelyNearSemigroups returns this sequence as a list of minimal generating sets, beginning with mingens S and ending with {1}.

For an Arf semigroup, every blowup in the sequence has minimal multiplicity (see isArf), and the sequence of multiplicities determines the Arf closure (see arfClosure):