PreprintCC BY-NC-SA 4.0arXiv
Chain Continuity and Scott-Closed Interpolation
Abstract
We prove that every chain-continuous dcpo is continuous and that its way-below relation is the relational square of its chain-below relation, answering a question of Erné. A subalgebra lattice obtained from a locally finite Jónsson algebra shows that the chain-below relation itself need not interpolate. For approximation by Scott-closed sets, we give a finite-upper-bound characterization on lower-set lattices and characterize interpolation by an extension property for directed lower sets. This yields a countably based, completely distributive algebraic lattice on which chain and directed approximation coincide, whereas Scott-closed approximation fails to interpolate, answering questions of Erné and of Ho and Zhao.
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BibTeX
@misc{chen2026chain,
title = {{Chain Continuity and Scott-Closed Interpolation}},
author = {Yuxu Chen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18692},
eprint = {2610.18692},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18692},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Chain Continuity and Scott-Closed Interpolation
AU - Yuxu Chen
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18692
UR - https://arxiv.org/abs/2610.18692
AB - We prove that every chain-continuous dcpo is continuous and that its way-below relation is the relational square of its chain-below relation, answering a question of Erné. A subalgebra lattice obtained from a locally finite Jónsson algebra shows that the chain-below relation itself need not interpolate. For approximation by Scott-closed sets, we give a finite-upper-bound characterization on lower-set lattices and characterize interpolation by an extension property for directed lower sets. This yields a countably based, completely distributive algebraic lattice on which chain and directed approximation coincide, whereas Scott-closed approximation fails to interpolate, answering questions of Erné and of Ho and Zhao.
ER -