Numerical semigroups whose fractions are of maximal embedding dimension

Each saturated (resp., Arf) numerical semigroup S has the property that each of its fractions (frac{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∈?, (S = frac{T}{k}) for infinitely many MED (resp., Arf; resp., saturated) numerical semigroups T. Let (mathcal{A}) (resp., (mathcal{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 (mathcal{A} =mathcal{C}_{1} subsetmathcal{C}_{2} subsetmathcal{C}_{3}subset ,cdots, subsetmathcal{F}), where, like (mathcal{A}) and (mathcal{F}), each (mathcal{C}_{n}) is stable under the formation of fractions.