On Numerical Semigroups Generated by Generalized Arithmetic Sequences#

Abstract

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 . Set B 0(S) = S, and for i ≥ 1, define B i (S) ≔ B(B i−1(S)). Similarly, set L 0(S) = S, and for i ≥ 1, define L i (S) ≔ L(L i−1(S)). These constructions define two finite ascending chains of numerical semigroups S = B 0(S) ⊆ B 1(S) ⊆ … ⊆ B β(S)(S) = ℕ0 and S = L 0(S) ⊆ L 1(S) ⊆ … ⊆ L λ(S)(S) = ℕ0. It has been shown that not all numerical semigroups S have the property that B i (S) ⊆ L i (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 B i (S) ⊆ L i (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.