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PreprintarXiv licencearXiv

A Symmetrization for Almost Symmetric Numerical Semigroups

  • Manuel Delgado
  • Meral Süer
  • Mehmet Yeşil
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Abstract

We characterize almost symmetric numerical semigroups in terms of their associated Young diagrams, showing that almost symmetry is equivalent to a simple numerical identity relating the genus, the type, and the number of small elements, and also to a combinatorial condition on the hook lengths of the diagram. We then introduce a symmetrization procedure that associates with each almost symmetric numerical semigroup a symmetric numerical set. This yields a map from the set of almost symmetric numerical semigroups to the set of symmetric numerical sets. We also present experimental data on the image of this map and on the proportion of its elements that are numerical semigroups. The results show, in particular, that the map is far from injective and that its image contains many numerical sets that are not semigroups.

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Cite

BibTeXRIS
BibTeX
@misc{delgado2026symmetrization,
  title = {{A Symmetrization for Almost Symmetric Numerical Semigroups}},
  author = {Manuel Delgado and Meral Süer and Mehmet Yeşil},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.14268},
  eprint = {2611.14268},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.14268},
}
RIS
TY  - UNPB
TI  - A Symmetrization for Almost Symmetric Numerical Semigroups
AU  - Manuel Delgado
AU  - Meral Süer
AU  - Mehmet Yeşil
PY  - 2026
DA  - 2026/11/10
PB  - arXiv
DO  - 10.48550/arxiv.2611.14268
UR  - https://arxiv.org/abs/2611.14268
AB  - We characterize almost symmetric numerical semigroups in terms of their associated Young diagrams, showing that almost symmetry is equivalent to a simple numerical identity relating the genus, the type, and the number of small elements, and also to a combinatorial condition on the hook lengths of the diagram. We then introduce a symmetrization procedure that associates with each almost symmetric numerical semigroup a symmetric numerical set. This yields a map from the set of almost symmetric numerical semigroups to the set of symmetric numerical sets. We also present experimental data on the image of this map and on the proportion of its elements that are numerical semigroups. The results show, in particular, that the map is far from injective and that its image contains many numerical sets that are not semigroups.
ER  -