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PreprintCC BY 4.0arXiv

Cσ-Unique Dcpos: Intrinsic Characterizations and Counterexamples

  • Yuxu Chen
  • Xulong He
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Abstract

A dcpo D is called Cσ-unique if for every dcpo Q, Γ(D)≅ Γ(Q) implies D≅ Q, where Γ(D) denotes the lattice of Scott-closed subsets of D. We give an intrinsic characterization of Cσ-uniqueness in terms of Skula-density and Scott closure, identifying precisely when a proper sub-dcpo can preserve the lattice of Scott-closed sets. Based on this characterization, we answer three open problems: (1) every power JI of Johnstone's dcpo is Cσ-unique; (2) Cσ-uniqueness is not preserved by binary products, even when both factors and their product are sober; and (3) a sober countable complete lattice need not be Cσ-unique, even when it is a frame.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026cσunique,
  title = {{$C_σ$-Unique Dcpos: Intrinsic Characterizations and Counterexamples}},
  author = {Yuxu Chen and Xulong He},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.40191},
  eprint = {2609.40191},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.40191},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - C_σ-Unique Dcpos: Intrinsic Characterizations and Counterexamples
AU  - Yuxu Chen
AU  - Xulong He
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.40191
UR  - https://arxiv.org/abs/2609.40191
AB  - A dcpo D is called C_σ-unique if for every dcpo Q, Γ(D)≅ Γ(Q) implies D≅ Q, where Γ(D) denotes the lattice of Scott-closed subsets of D. We give an intrinsic characterization of C_σ-uniqueness in terms of Skula-density and Scott closure, identifying precisely when a proper sub-dcpo can preserve the lattice of Scott-closed sets. Based on this characterization, we answer three open problems: (1) every power J^I of Johnstone's dcpo is C_σ-unique; (2) C_σ-uniqueness is not preserved by binary products, even when both factors and their product are sober; and (3) a sober countable complete lattice need not be C_σ-unique, even when it is a frame.
ER  -