On Triply-generated Telescopic Semigroups and Chains of Semigroups

Given a numerical semigroup S = 〈a1, a2, . . . , aν〉 in canonical form, let M(S) := S \ {0}. Define associated numerical semigroups B(S) := {x ∈ N0 : x + M(S) ⊆ M(S)} and L(S) := 〈a1, a2 − a1, . . . , aν − a1〉 . Set B0(S) = S, and for i ≥ 1, define Bi(S) := B(Bi−1(S)). Similarly, set L0(S) = S, and for i ≥ 1, define Li(S) := L(Li−1(S)). These constructions define finite ascending chains of semigroups S = B0(S) ⊆ B1(S) ⊆ · · · ⊆ Bβ(S)(S) = N0 and S = L0(S) ⊆ L1(S) ⊆ · · · ⊆ Lλ(S)(S) = N0. It is shown that if S is a triply-generated telescopic semigroup, then Bj(S) = L1(S) for some j, 1 ≤ j ≤ β(S). From this, it follows that certain triplygenerated telescopic semigroups S satisfy Bi(S) ⊆ Li(S) for all 0 ≤ i ≤ β(S).