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On comparing two chains of numerical semigroups and detecting Arf semigroups

By David E. Dobbs and Gretchen L. Matthews

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 {Bi(S) : 0 ≤ i ≤ β(S)} and {Li(S) : 0 ≤ i ≤ λ(S)} of semigroups by B0(S): = S =: L0(S), B β(S)(S): = N =: L λ(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)

Topics: Numerical semigroup, Lipman semigroup of an ideal, maximal embedding dimension, symmetric semigroup, Arf semigroup. 1991 Mathematics Subject Classification
Year: 2001
OAI identifier: oai:CiteSeerX.psu:10.1.1.134.9014
Provided by: CiteSeerX
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