Scott products and sobriety of countable meet-continuous dcpos
Abstract
We study Scott-product compatibility and sobriety for countable meet-continuous directed-complete partial orders (dcpos). We construct a countable meet-continuous dcpo ℛ whose Scott space is well-filtered but not sober, although the Scott topology on every finite power is the product of the factor Scott topologies. The ideals of ℛ admit a unique finite-parameter description. We also construct a countable meet-continuous dcpo ℍ with a greatest element whose Scott space is coherent and well-filtered, but whose Scott square has a topology strictly finer than the ordinary product topology. These constructions answer negatively two questions about countable meet-continuous dcpos. On the positive side, we prove that a meet-continuous L-dcpo, meaning a dcpo whose principal ideals are complete lattices, is Scott sober whenever its Scott square carries the ordinary product topology. This answers Jia's core-compactness question affirmatively for L-dcpos without any countability assumption. Finally, we characterize well-filteredness of L-dcpos by boundedness of closed Rudin sets and formulate related open questions.
Related
BibTeX
@misc{xu2026scott,
title = {{Scott products and sobriety of countable meet-continuous dcpos}},
author = {Xiaoquan Xu and Wei Ji},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02642},
eprint = {2610.02642},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02642},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Scott products and sobriety of countable meet-continuous dcpos
AU - Xiaoquan Xu
AU - Wei Ji
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02642
UR - https://arxiv.org/abs/2610.02642
AB - We study Scott-product compatibility and sobriety for countable meet-continuous directed-complete partial orders (dcpos). We construct a countable meet-continuous dcpo ℛ whose Scott space is well-filtered but not sober, although the Scott topology on every finite power is the product of the factor Scott topologies. The ideals of ℛ admit a unique finite-parameter description. We also construct a countable meet-continuous dcpo ℍ with a greatest element whose Scott space is coherent and well-filtered, but whose Scott square has a topology strictly finer than the ordinary product topology. These constructions answer negatively two questions about countable meet-continuous dcpos. On the positive side, we prove that a meet-continuous L-dcpo, meaning a dcpo whose principal ideals are complete lattices, is Scott sober whenever its Scott square carries the ordinary product topology. This answers Jia's core-compactness question affirmatively for L-dcpos without any countability assumption. Finally, we characterize well-filteredness of L-dcpos by boundedness of closed Rudin sets and formulate related open questions.
ER -