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Supersymmetric gaps of a numerical semigroup with two generators

Patricio Almirón, Julio José Moyano-Fernández

TL;DR

The paper investigates numerical semigroups with two generators by introducing supersymmetric and self-symmetric gaps relative to the Wilf number. It encodes gap information as lattice points and uses a lattice-path/semimodule framework to define the Wilf number and its zero-set properties; the central result shows that the combined gap sets $SG∪SSG$ determine the entire semigroup $Γ=⟨α,β⟩$ via a polyomino game and symmetry operations. It also compares this gap-based generating system with the classical fundamental gaps, derives parity-based cardinality bounds, and discusses limitations and extensions to higher embedding dimensions. Overall, the work provides a compact, symmetry-driven representation that determines the semigroup and offers new avenues toward understanding Wilf’s conjecture. The results highlight a rich geometric-combinatorial structure in the gap set that complements existing representations such as fundamental gaps.

Abstract

In this paper we introduce the new concepts of supersymmetric and self-symmetric gaps of a numerical semigroup with two generators. Those concepts are based on certain symmetries of the gaps of the semigroup with respect to their Wilf number. We prove that the set of supersymmetric and self-symmetric gaps completely determines the semigroup and we compare this set with the fundamental gaps of the semigroup.

Supersymmetric gaps of a numerical semigroup with two generators

TL;DR

The paper investigates numerical semigroups with two generators by introducing supersymmetric and self-symmetric gaps relative to the Wilf number. It encodes gap information as lattice points and uses a lattice-path/semimodule framework to define the Wilf number and its zero-set properties; the central result shows that the combined gap sets determine the entire semigroup via a polyomino game and symmetry operations. It also compares this gap-based generating system with the classical fundamental gaps, derives parity-based cardinality bounds, and discusses limitations and extensions to higher embedding dimensions. Overall, the work provides a compact, symmetry-driven representation that determines the semigroup and offers new avenues toward understanding Wilf’s conjecture. The results highlight a rich geometric-combinatorial structure in the gap set that complements existing representations such as fundamental gaps.

Abstract

In this paper we introduce the new concepts of supersymmetric and self-symmetric gaps of a numerical semigroup with two generators. Those concepts are based on certain symmetries of the gaps of the semigroup with respect to their Wilf number. We prove that the set of supersymmetric and self-symmetric gaps completely determines the semigroup and we compare this set with the fundamental gaps of the semigroup.

Paper Structure

This paper contains 7 sections, 15 theorems, 55 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Lattice paths and semimodules over a numerical semigroup
  3. Wilf number of a gap of a numerical semigroup
  4. Supersymmetry of the gap set with respect to the Wilf number
  5. Determinacy of the semigroup
  6. Fundamental gaps vs supersymmetric gaps and self-symmetric gaps
  7. Remarks on the concepts of supersymmetric and self-symmetric gaps

Key Result

Proposition 2.2

where $h_1,\ldots , h_{n-1}$ are gaps of $\Gamma$, $h_0,h_n \leq \alpha \beta$, and

Figures (10)

  • Figure 2.1: Lattice path for the $\langle 5,7 \rangle$-lean set $[0,9,11,8]$.
  • Figure 4.1: Lattice representation of the gap set $\mathbb{N}\setminus\Gamma$. The shaded set is $\mathsf{SG}$ and the striped one is $\mathsf{SSG}$.
  • Figure 4.2: The sets $\mathcal{T}_u\cup s_{\alpha}(\mathcal{T}_u)$ (starred) and $\mathcal{T}_r\cup s_{\beta}(\mathcal{T}_r)$ (shaded).
  • Figure 4.3: Sets $\mathcal{T}_u\cup s_{\alpha}(\mathcal{T}_u)$ (shaded), $\mathsf{SSG}$ (starred), and $\mathcal{T}_r\cup s_{\beta}(\mathcal{T}_r)$ (dotted).
  • Figure 4.4: The set $\mathsf{SG}$ (shaded).
  • ...and 5 more figures

Theorems & Definitions (37)

  • Conjecture 1.1: Wilf conjecture
  • Definition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 27 more