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.

Key Words:

• Numerical semigroup
• Dual of a semigroup
• Lipman semigroup
• Arf semigroup
• Generalized arithmetic sequence

1991 Mathematics Subject Classifications:

• Primary 20M99
• Secondary 20M14, 20M12

Acknowledgments

Notes

^#Communicated by I. Swanson.