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Infinite chains in the tree of numerical semigroups

Mariana Rosas-Ribeiro, Maria Bras-Amorós

Abstract

One major problem in the study of numerical semigroups is determining the growth of the semigroup tree. In the present work, infinite chains of numerical semigroups in the semigroup tree, firstly introduced by Bras-Amorós and Bulygin (Semigroup Forum, 79:561--574, 2009), are studied. Computational results show that these chains are rare, but without them the tree would not be infinite. It is proved that for each genus $g\geq 5$ there are more semigroups of that genus not belonging to infinite chains than semigroups belonging. Bras-Amorós and Bulygin (Semigroup Forum, 79:561--574, 2009) presented a characterization of the semigroups that belong to infinite chains in terms of the coprimality of the left elements of the semigroup as well as a result on the cardinality of the set of infinite chains to which a numerical semigroup belongs in terms of the primality of the greatest common divisor of these left elements. We revisit these results and fix an imprecision on the cardinality of the set of infinite chains to which a semigroup belongs in the case when the greatest common divisor of the left elements is a prime number. We then look at infinite chains in subtrees with fixed multiplicity. When the multiplicity is a prime number there is only one infinite chain in the tree of semigroups with such multiplicity. When the multiplicity is $4$ or $6$ we prove a self-replication behavior in the subtree and prove a formula for the number of semigroups in infinite chains of a given genus and multiplicity $4$ and $6$, respectively.

Infinite chains in the tree of numerical semigroups

Abstract

One major problem in the study of numerical semigroups is determining the growth of the semigroup tree. In the present work, infinite chains of numerical semigroups in the semigroup tree, firstly introduced by Bras-Amorós and Bulygin (Semigroup Forum, 79:561--574, 2009), are studied. Computational results show that these chains are rare, but without them the tree would not be infinite. It is proved that for each genus there are more semigroups of that genus not belonging to infinite chains than semigroups belonging. Bras-Amorós and Bulygin (Semigroup Forum, 79:561--574, 2009) presented a characterization of the semigroups that belong to infinite chains in terms of the coprimality of the left elements of the semigroup as well as a result on the cardinality of the set of infinite chains to which a numerical semigroup belongs in terms of the primality of the greatest common divisor of these left elements. We revisit these results and fix an imprecision on the cardinality of the set of infinite chains to which a semigroup belongs in the case when the greatest common divisor of the left elements is a prime number. We then look at infinite chains in subtrees with fixed multiplicity. When the multiplicity is a prime number there is only one infinite chain in the tree of semigroups with such multiplicity. When the multiplicity is or we prove a self-replication behavior in the subtree and prove a formula for the number of semigroups in infinite chains of a given genus and multiplicity and , respectively.
Paper Structure (13 sections, 24 theorems, 20 equations, 5 figures, 1 table)

This paper contains 13 sections, 24 theorems, 20 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Infinite chains
  4. Minority of semigroups in infinite chains
  5. Fixing the multiplicity
  6. Prime multiplicity
  7. Multiplicity 4
  8. Multiplicity 6
  9. The tree $\tau_n$ and its replications
  10. Outside the replications of $\tau_n$
  11. Growth of the tree $\upsilon$
  12. Open question
  13. Acknowledgments

Key Result

Lemma 2.1

If $\lambda_{i_1}<\lambda_{i_2}<\dots<\lambda_{i_n}$ are the effective generators of a non-ordinary numerical semigroup $\Lambda$, then the effective generators of $\Lambda\setminus\{\lambda_{i_j}\}$ are either $\lambda_{i_{j+1}},\dots,\lambda_{i_n}$ or $\lambda_{i_{j+1}},\dots,\lambda_{i_n},\lambda

Figures (5)

  • Figure 1: Tree of numerical semigroups with the nodes of level at most 6.
  • Figure 2: Tree structure of numerical semigroups of multiplicity $4$ that are in an infinite chain, from genus $3$ to $5$.
  • Figure 3: Tree of numerical semigroups in infinite chains, with multiplicity $4$, from genus $3$ up to $41$.
  • Figure 4: Finite tree $\tau_n$
  • Figure 5: Numerical semigroups of multiplicity $6$ in infinite chains, from genus $5$ up to genus $29$. The subtree $\tau_3$ is highlighted in blue. The parts of the chains $\eta_3$ and $\zeta_3$ of genus larger than or equal to $19$ are highlighted in green.

Theorems & Definitions (48)

  • Lemma 2.1: m-towards, Lemma 1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Remark 3.3
  • Theorem 3.4
  • ...and 38 more