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Generalized Numerical semigroups up to isomorphism

Carmelo Cisto, Gioia Failla, Francesco Navarra

TL;DR

This work classifies generalized numerical semigroups (GNS) up to isomorphism by showing that any isomorphism between GNSs in $\mathbb{N}^d$ is induced by a coordinate permutation in $P_d$. It then develops two tree-based frameworks and representative-selection strategies to enumerate non-isomorphic GNSs of fixed genus, including equivariant GNSs and the $\operatorname{R}_{\preceq}$ and $\mathcal{O}$-based constructions. The authors provide concrete algorithms and computational data, establishing that the number of isomorphism classes $N_{g,d}$ equals $N_{g,g}$ for $d\ge g$ and presenting extensive genus-wise counts. The results advance practical enumeration of GNSs and deepen connections with affine semigroups, Apéry sets, and PF/SG structures, with implications for combinatorial and computational aspects of semigroup theory.

Abstract

A generalized numerical semigroup is a submonoid $S$ of $\mathbb{N}^d$ with finite complement in it. We characterize isomorphisms between these monoids in terms of permutation of coordinates. Considering the equivalence relation that identifies the monoids obtained by the action of a permutation and establishing a criterion to select a representative from each equivalence class, we define some procedures for generating the set of all generalized numerical semigroups of given genus up to isomorphism. Finally, we present computational data and explore properties related to the number of generalized numerical semigroups of a given genus up to isomorphism.

Generalized Numerical semigroups up to isomorphism

TL;DR

This work classifies generalized numerical semigroups (GNS) up to isomorphism by showing that any isomorphism between GNSs in is induced by a coordinate permutation in . It then develops two tree-based frameworks and representative-selection strategies to enumerate non-isomorphic GNSs of fixed genus, including equivariant GNSs and the and -based constructions. The authors provide concrete algorithms and computational data, establishing that the number of isomorphism classes equals for and presenting extensive genus-wise counts. The results advance practical enumeration of GNSs and deepen connections with affine semigroups, Apéry sets, and PF/SG structures, with implications for combinatorial and computational aspects of semigroup theory.

Abstract

A generalized numerical semigroup is a submonoid of with finite complement in it. We characterize isomorphisms between these monoids in terms of permutation of coordinates. Considering the equivalence relation that identifies the monoids obtained by the action of a permutation and establishing a criterion to select a representative from each equivalence class, we define some procedures for generating the set of all generalized numerical semigroups of given genus up to isomorphism. Finally, we present computational data and explore properties related to the number of generalized numerical semigroups of a given genus up to isomorphism.
Paper Structure (8 sections, 25 theorems, 8 equations, 3 figures, 1 table, 2 algorithms)

This paper contains 8 sections, 25 theorems, 8 equations, 3 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and notations
  3. Permutations and isomorphisms between GNSs
  4. Equivariant generalized numerical semigroups
  5. Computing non-isomorphic generalized numerical semigroups of fixed genus
  6. Representatives and a first construction
  7. A different construction for producing $\operatorname{R}_{\preceq}(\mathcal{S}_{g,d})$
  8. On the number of GNSs up to isomorphism and computational data

Key Result

Proposition 2.4

Let $S\subseteq \mathbb{N}^d$ be a GNS and $\mathbf{h}\in \operatorname{H}(S)$. Then $S\cup \{\mathbf{h}\}$ is a GNS if and only if $\mathbf{h}\in \operatorname{SG}(S)$.

Figures (3)

  • Figure 1: Some branches of the tree $\mathcal{G}^2_\preceq$.
  • Figure 2: Some branches of $\mathcal{T}_{2,\preceq}$ and $\mathcal{T}_{d,\preceq}^{\operatorname{R}}$ where $\preceq$ is the lexicographic order, with reference to Example \ref{['exa-tree1']}.
  • Figure 3: The tree $\mathcal{K}_{3,2,\preceq}^{\operatorname{R}}$, where $\preceq$ is the lexicographic order, with reference to Example \ref{['exa:tree2']}

Theorems & Definitions (65)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Proposition 2.4
  • Remark 3.1
  • Definition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • proof
  • Lemma 3.5
  • ...and 55 more