On comparing two chains of numerical semigroups and detecting Arf semigroups

If T is a numerical semigroup with maximal ideal N , define associated semigroups B(T ) := (N −N) and L(T ) = ∪{(hN −hN) : h ≥ 1}. If S is a numerical semigroup, define strictly increasing finite sequences {Bi(S) : 0 ≤ i ≤ β(S)} and {Li(S) : 0 ≤ i ≤ λ(S)} of semigroups by B0(S) := S =: L0(S), Bβ(S)(S) := N =: Lλ(S)(S), Bi+1(S) := B(Bi(S)) for 0 < i < β(S), Li+1(S) := L(Li(S)) for 0 < i < λ(S). It is shown, contrary to recent claims and conjectures, that B2(S) need not be a subset of L2(S) and that β(S)−λ(S) can be any preassigned integer. On the other hand, B2(S) ⊆ L2(S) in each of the following cases: S is symmetric; S has maximal embedding dimension; S has embedding dimension e(S) ≤ 3. Moreover, if either e(S) = 2 or S is pseudo-symmetric of maximal embedding dimension, then Bi(S) ⊆ Li(S) for each i, 0 ≤ i ≤ λ(S). For each integer n ≥ 2, an example is given of a (necessarily non-Arf) semigroup S such that β(S) = λ(S) = n, Bi(S) = Li(S) for all 0 ≤ i ≤ n− 2, and Bn−1(S) $ Ln−1(S).