On Numerical Semigroups Generated by Generalized Arithmetic Sequences.

Given a numerical semigroup S. let M(S) = S\{0} and (IM(S) - IM(S)) - {x ϵ No : x · IM(S) ⊆ IM(S)}. Define associated numerical semigroups B(S) := (M(S) - M(S)) and L(S) := Ut=i∞ (IM(S) - IM(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) ⊆ … ⊆ BβS(S) - N0 and S - L0(S) ⊆ L1(S) ⊆ … ⊆ Lλ(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 from 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. [ABSTRACT FROM AUTHOR]