On Numerical Semigroups Generated by Generalized Arithmetic Sequences

Given a numerical semigroup S, let M(S) = S \{0} and (lM(S)− lM(S)) = {x ∈ N0 : x + lM(S) ⊆ lM(S)}. Define associated numerical semigroups B(S) := (M(S)−M(S)) and L(S) := ∪l=1(lM(S)− lM(S)). 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 two finite ascending chains of numerical semigroups S = B0(S) ⊆ B1(S) ⊆ · · · ⊆ Bβ(S)(S) = N0 and S = L0(S) ⊆ L1(S) ⊆ · · · ⊆ Lλ(S)(S) = N0. It has been shown that not all numerical semigroups S have the property that Bi(S) ⊆ Li(S) for all i ≥ 0. In this paper, we prove that if S is a numerical semigroup with a set of generators that form a generalized arithmetic sequence, then Bi(S) ⊆ Li(S) for all i ≥ 0. Moreover, we see that this containment is not necessarily satisfied if a set of generators of S form an almost arithmetic sequence. In addition, we characterize numerical semigroups generated by generalized arithmetic sequences that satisfy other semigroup properties, such as symmetric, pseudo-symmetric, and Arf.