An algorithm to compute the primitive elements of an embedding dimension three numerical semigroup

Abstract

We give an algorithm to compute the set of primitive elements for an embedding dimension three numerical semigroups. We show how we use this procedure in the study of the construction of L-shapes and the tame degree of the semigroup.

Delta sets for nonsymmetric numerical semigroups with embedding dimension three

“…A natural way to study ∆(S) passes through a better understanding of M . This is because δ ∈ ∆(S) if and only if (1) there exists x, y ∈ Z(s) for some s ∈ S, such that |x| > |y| and δ = |x| − |y| (= |x − y|), and (2) there is no z ∈ Z(s) such that |x| > |z| > |y|. The first condition relies on M and for the second we introduce the concept of Bézout couples.…”

“…In practice, when we are only interested in the Delta set, we do not need to keep track of the Bézout couples, just the integers appearing in the greatest common divisor computation. gap> deltasetnsembdim3(1407, 26962, 35413);time; [ 1,2,3,4,7,10,13,23,33,43,76,109,142,251, 393 ] 1 The time is in milliseconds, that is, it takes 1 millisecond to compute ∆(S). The current procedure DeltaSetOfNumericalSemigroup in numericalsgps executed with this example was stopped after one day without an output.…”

“…gap> s:=NumericalSemigroup(101,301,510);; gap> DeltaSetOfNumericalSemigroup(s);time; [ 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17, 18 ] 10271 gap> deltasetnsembdim3(101,301,510);time; [ 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 (we remove −v if v is already in the basis). When looking for v 1 we need to search for the elements (a, b, c) ∈ G with a = 0 and bc ≥ 0: {(4, −1, −1), (5, −3, 0), (7, 0, −3)}.…”

On the number of L-shapes in embedding dimension four numerical semigroups

“…In order to look for primitive elements, we developed an algorithm in [2] that gave us some light to find the family of semigroups that we present in this paper.…”

“…. , j k ) ∈ N k fulfilling j 1 s 1 + · · · + j k s k ≡ i 1 s 1 + · · · + i k s k (mod N ), (2) if v = [[j 1 , . .…”