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PreprintCC BY 4.0arXiv

Kobayashi's problem for power semigroups of inverse semigroups

  • Shuolin Liu
  • Salvatore Tringali
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Abstract

For a semigroup S (written multiplicatively), let 𝒫(S) denote the semigroup of all non-empty subsets of S under setwise multiplication. The finitary power semigroup 𝒫fin(S) is the subsemigroup of 𝒫(S) consisting of all non-empty finite subsets of S. Kobayashi's problem asks for which classes π’ͺ of semigroups every isomorphism 𝒫(H) β†’ 𝒫(K) with H, K ∈ π’ͺ maps singletons bijectively onto singletons and hence induces an isomorphism H β†’ K. We give an affirmative answer for inverse semigroups, extending previous results for groups, semilattices, and, more generally, Clifford semigroups. We also establish the corresponding rigidity theorem for finitary power semigroups. To this end, we develop a new framework that provides a common approach to both settings.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026kobayashis,
  title = {{Kobayashi's problem for power semigroups of inverse semigroups}},
  author = {Shuolin Liu and Salvatore Tringali},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29525},
  eprint = {2610.29525},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29525},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Kobayashi's problem for power semigroups of inverse semigroups
AU  - Shuolin Liu
AU  - Salvatore Tringali
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29525
UR  - https://arxiv.org/abs/2610.29525
AB  - For a semigroup S (written multiplicatively), let 𝒫(S) denote the semigroup of all non-empty subsets of S under setwise multiplication. The finitary power semigroup 𝒫_fin(S) is the subsemigroup of 𝒫(S) consisting of all non-empty finite subsets of S. Kobayashi's problem asks for which classes π’ͺ of semigroups every isomorphism 𝒫(H) β†’ 𝒫(K) with H, K ∈ π’ͺ maps singletons bijectively onto singletons and hence induces an isomorphism H β†’ K. We give an affirmative answer for inverse semigroups, extending previous results for groups, semilattices, and, more generally, Clifford semigroups. We also establish the corresponding rigidity theorem for finitary power semigroups. To this end, we develop a new framework that provides a common approach to both settings.
ER  -