The doubles of one half of a numerical semigroup

Let FF be the set of all numerical semigroups and d   a positive integer. Define a mapping fd:F→Ffd:F→F for every numerical semigroup S   by fd(S)=Sd. In this paper, we show that the equivalence class (S)kerfd(S)kerfd contains only finite maximal elements, but does not contain minimal element for every integer d≥2d≥2. Moreover, we study the subset of (S)kerf2(S)kerf2 whose elements contain S   and the subset of (S)kerf2(S)kerf2 whose elements are contained in S. In particular, we study the case of S has embedding dimension two.