On comparing two chains of numerical semigroups and detecting arf semigroups

Abstract

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 { B i (S) : 0 ≤ i ≤β (S) } and { L i (S) : 0 ≤ i ≤λ (S) } of semigroups by B 0 (S):=S=:L 0 (S) , B β (S) (S):= \Bbb N =: L λ (S) (S) , B i+1 (S):=B(B i (S)) for 0<i< β (S) , L i+1 (S):=L(L i (S)) for 0<i< λ (S) . It is shown, contrary to recent claims and conjectures, that B 2 (S) need not be a subset of L 2 (S) and that β (S) - λ (S) can be any preassigned integer. On the other hand, B 2 (S) ⊆ L 2 (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 B i (S) ⊆ L i (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 , B i (S) = L i (S) for all 0 ≤ i ≤ n-2 , and B n-1 (S) \subsetneqq L n-1 (S) .