Numerical semigroups whose fractions are of maximal embedding dimension

Each saturated (resp., Arf) numerical semigroup S has the property that each of its fractions S/k is saturated (resp., Arf), but the property of being of maximal embedding dimension (MED) is not stable under formation of fractions. If S is a numerical semigroup, then S is MED (resp., Arf; resp., saturated) if and only if, for each 2 <= k is an element of N, S = T/k for infinitely many MED (resp., Arf; resp., saturated) numerical semigroups T. Let A (resp., F) be the class of Arf numerical semigroups (resp., of numerical semigroups each of whose fractions is of maximal embedding dimension). Then there exists an infinite strictly ascending chain A = C-1 subset of C-2 subset of C-3 subset of ... subset of F, where, like A and F, each C-n is stable under the formation of fractions.